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# Benefits/Drawbacks of highly custamizable games.

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Benefits/Drawbacks of highly custamizable games. -or- Extreme Ability class gap. -or- Where exactly does cheating begin?

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No. Quake 3 does not cross over any lines. How is customizing your autoexec file cheating? It''s not. They allow you to alter your setup. Yes, using an aim bot or wall hack is cheating. But not modifying some client side variables.

In my autoexec file, my brightness is at 100%, I have different sensitivities for each weapon (higher senses for rocket launcher, etc, lower ones for railgun), I have cl_mouseAccel on 0.0225, I have poly levels on the lowest possible level, r_picmip is on 5..etc.

I like it that way. It''s how I want my copy of the game . It''s customized by me, to work for me . There is no crossing of any line.

Everyone knows graphics are not everything. But your post would say otherwise. You''re saying people are getting too much into their frag count, rather than enjoying the "wow aspect" (visual aspect). The game was meant to be PLAYABLE. To have great gameplay. Wanting a higher frag score is PART of that. The point of the game is not to oogle at the graphics, it''s to have some damned competitive fun.

To simply answer your question again: No, Quake 3 is not crossing any lines.

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I think that he is not trying to lean in one direction or another. Just asking the question: anyway. I think that when someone uses a script to make themselves jump/duck, then fine, because I can do it manually w/ no problem and I kinda pride myself in being able to do it. Someone may argue that when you do it manually sometimes you might mess up, but the jump/duck script NEVER will, except if the person hits the scripted button to early or late or whatever. So you might argue that the script takes the human element of error occuring: consider if you are in a messy situation and you are bit nervouse, you might mess up a manual jump/duck, but the script will make it so that your character acts inhumanly perfect in a messy situation, even though you yourself are freaking out.. see what I mean?
Now a jump/duck just reduces pushing 2 buttons to 1, but what about 3 or more seperate body maneuvers in one scripted button? To me that is getting stupid, because you multiply the argument that I listed above.

I personally play manually, and I do VERY well, both at Q1, and CS, Tribes, and I pride myself in being able to keep my cool and still be able to perform delicate, skilled moves, not just hit a button and let the computer do it for me like a little no-skill chump!

Thats my take on it.

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I''d probably start a war with this comment, but the Quake series weren''t so much games as they were engine demos. In other words, Carmack makes more money from selling his engine to other game making companies then he does from actual sale of the game. Not that the second is an insignificant number, just when programming the engine, that was the primary thought on his mind. When he finished the engine, it was passed on to the development team and his active participation in Quake reduced to drawing design docs and giving pep talks. With this in mind, its no wonder that Id arranges quakefest. To bring the gamers together and put a small message out to other companies, "look what my engine will get you."

As to customization in a game, this is totally at the discretion of the developer. They have control over what the player can be doing. if they want you to cheat, they leave in quicksave and cheat systems. As a developer, if you don''t want this, then don''t include it. On the other hand, if your a gamer, my only advice is to stop buying the games that you don''t like.

-> Will Bubel
-> Machine wash cold, tumble dry.

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dont give me that shit that custmisation isnt a form of cheating. For starters, lets ask ourselves the purpose of the wallhack, its to see things/people where by default you couldnt see. Now take CS for example, if i leave my brightness at the default, there is shadows and everything that people can hide in, you cant see people in these dark places and that is how the game was designed. If i whack my brightness up, i can everything now, no darkness. So anyone who used to hide here and get me is no longer hidden and i can attack them.

Hang on a sec, does this not sound familar to the purpose of the wallhack, changing the brightness allows you to see things in places by default you couldnt. Its exactly the same purpose and is therefore CHEATING.

And when you get onto scripts, its just pathetic, the moment the gameplay is taken out of your hands and is placed upon the computer (aimbot etc) thats cheating, and thats exactly what jump and shoot scripts do. I dont care if you think its customising the game to how you want it, what a copout. Hell ill just go customise the game to how id like it, wallhack, aimbot YEAH that sounds fair, im not cheating, its just customising the game as you said.

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Anonymous, you are a fool. You obviously have no idea what a wall hack or aim bot actually does.

A wall hack allows you to see where other players are no matter where they are. Some wallhacks draw a little sprite on your screen over an enemy model, and you'll see that sprite running, even if there's a wall in front of you.

An aim bot does just that, it uses its native program code to control your aim for you. One particular aim bot allows you to bind the command "+autoaim" to a key (a command that IT CREATED, and ONLY IT recognizes), and as long as you hold the key, the program will automatically snap your aim to the nearest opponent, and fire.

None of these can be done simply by modifying a view client side variables. They all rely on the programming of the author/s to do this job for them.

Turning up the brightness of your screen is not cheating. Where do you even get the grounds for saying this? The developers GIVE you the ability to do it, it's YOUR choice to change the brightness. Everyone can do it if they want to. I have my brightness all the way up; now I can see more clearly. Am I cheating? No. I'm using an option that id software gave me. Other players can too. And most of them do .

You must not be a real FPS player if you think that all of these games must be played exactly how they were shipped to you, with all settings on default. This is such bullcrap. Any real player customizes his game to work best for him. This is the purpose of having your own configuration file .

[edited by - munkie on July 26, 2002 12:16:15 AM]

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The one problem with player customizations is the possibility of exploitation(which brightness and macros do to a degree) and these exploitations being used to gain an advantage over the other players because they don''t know or are limited from doing the same thing. Even in the earliest days of (popular)online games like MUDs and such, people have put together macros to heal themselves or cast spells.

So the only things a developer should have to worry about in this regard, really, is making sure that whatever advantage gained is not overwhelmingly great, and that it''s not an extremely complicated process for a new player to reach the same level.

Making the world furry one post at a time

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quote:
Original post by munkie
Everyone knows graphics are not everything. But your post would say otherwise. You''re saying people are getting too much into their frag count, rather than enjoying the "wow aspect" (visual aspect). The game was meant to be PLAYABLE. To have great gameplay. Wanting a higher frag score is PART of that. The point of the game is not to oogle at the graphics, it''s to have some damned competitive fun.

You''re missing the point of the post completely. I did not state opinions saying graphics are everything. I merely commented on an increased focus on tweaked gameplay leading to an inversely proportional focus on graphics. The intent, which I thought was clear, was to discuss the merits and weaknesses of such designs, from the standpoints of both gamers and developers.

This then led to two concurrent discussions, the first being on whether this discourages newbies from trying to compete with players who have been playing for some time, or are familiar with a previous version, etc. If you had never played any quake game before, do you think you would have a lasting good time on any public multi-player server right now?

The second discussion revolves around where to draw the line for the definition of cheating in a multi player game. If merely changing client side variables is ok, why do some servers dissllow certain variables and limit others'' thresholds? Something to think about.

quote:
To simply answer your question again: No, Quake 3 is not crossing any lines.

I understand your point. However, Q3 is just a tangible example that many of us have experience with. The real question remains a comparison of the benefits/drawbacks of highly customizable games, and this should be further elaborated to clearly intend multi-player games.

--OctDev

The Tyr project is coming...

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quote:
Original post by TheEnderBean
I think that he is not trying to lean in one direction or another. Just asking the question:

Exactly.

quote:
I think that when someone uses a script to make themselves jump/duck, then fine, because I can do it manually w/ no problem and I kinda pride myself in being able to do it.

This can lead to an interesting situation that I hadn''t thought about before. Include base scripts with the shipping title. Now the newbies have these scripts and are made aware of them. Then, the trully elite are encouraged to perform these feats without the use of scripts. I have never even considered implementation before, but perhaps adding a % error to all scripted (multi-button) commands, so that trully perfect and consistent execution can only be achieved by masters who do it all manually. That...would...be...elite!

--OctDev

The Tyr project is coming...

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quote:
Original post by Inmate2993
I''d probably start a war with this comment, but the Quake series weren''t so much games as they were engine demos.

Agreed in some respects. Perhaps I seemed overly focused on quake in my original post. I meant it only as an example to help clarify the generic situation.

quote:
As to customization in a game, this is totally at the discretion of the developer.

I am curious. If you were developing a mp game, and you had the choice between the two, and you had 1000 top notch programmers working for you so it made NO difference either way (perfect world sceneario) which would you choose and why?

--OctDev

The Tyr project is coming...

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quote:
Original post by Anonymous Poster
dont give me that shit that custmisation isnt a form of cheating...Its exactly the same purpose and is therefore CHEATING.

And when you get onto scripts, its just pathetic, the moment the gameplay is taken out of your hands and is placed upon the computer (aimbot etc) thats cheating...

And this was (sorta) the opinion that I wanted to hear. I think in a single player game, the game has been designed around some sort of story and customization should not be as accessible. However, in typical mp environments, stories get thrown out the window in favor of killing each other. People can tweak their monitor settings outside the game. Shouldn''t it be accessible from within the game so as to keep from discouraging players with dark monitors from hassling with it every time they enter and leave the game?

You don''t like the additional features of being able to completely customize your gaming interface?

And what if someone is on a laggy connection? Shouldn''t they be allowed to tweak things to even up the ingame fighting?

--OctDev

The Tyr project is coming...

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quote:
Original post by munkie
A wall hack...
An aim bot...

None of these can be done simply by modifying a view client side variables. They all rely on the programming of the author/s to do this job for them.[quote]

So as long as you don''t supply any new code, it is not cheating then? Any allowable modfication of client side variables is ok though?

[quote]Turning up the brightness of your screen is not cheating.

Agreed.

quote:
You must not be a real FPS player if you think that all of these games must be played exactly how they were shipped to you, with all settings on default. This is such bullcrap. Any real player customizes his game to work best for him. This is the purpose of having your own configuration file .

What about the situation where someone is completely new to the FPS genre?

--OctDev

The Tyr project is coming...

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quote:
Original post by RTF
The one problem with player customizations is the possibility of exploitation(which brightness and macros do to a degree) and these exploitations being used to gain an advantage over the other players because they don''t know or are limited from doing the same thing...
So the only things a developer should have to worry about in this regard, really, is making sure that whatever advantage gained is not overwhelmingly great, and that it''s not an extremely complicated process for a new player to reach the same level.

Agreed. But how do you go about making it easily accessible for new players. So far, I think that my earlier thought of providing scripts but making scripts behave poorly by default is the best solution. Your thoughts?

--OctDev

The Tyr project is coming...

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The problems with inconsistent execution of something that could be done manually are these:

1. Some things, like brightness and sniper scripts, would stay unaffected.

2. Players feel like it''s a cop-out on the part of the developer that they deliberately made the game "harder" for them. They will try to find ways around any kind of protection against script exploiting, and it''s very likely that they''ll succeed before a few patches are released.

My own opinion is to give everyone the "keys to the kingdom," if you will, right from the in-game menus, and otherwise completely eliminate anything you don''t want to see in your game(bunny hopping, duckjumps etc.) since anything else will lead to animosity between players:

"That script''s a cheat"
"No it''s not"
"Yes it is cheater"
"STFU n00b"

The only way you can keep this kind of dissatisfaction down is by setting out as clearly as possible what you are allowed to do, and limit the rest completely.

Making the world furry one post at a time

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The intended use of client side variables is, or at least should be, to make the game playable, for instance by increasing the brightness on for someone with a darker monitor. Using them is not cheating per se, but I think that as soon as you use them to give yourself an advantage (eg turning brightness way up to see into shadows) you are entering a gray area if not cheating outright. Just because they''re there and permitted to use, does that make using them right? Should they be there?

People in quake 1 went to ridiculous lengths to give themselves an advantage in online play. They turned off texturing and lighting, made water invisible, used modified models, turned off head bobbing, changed the crosshair, used skins that blended into the background, they even turned up their FOV so they had a wider view angle and could see people beside/behind them. From what I remember most (maybe all) of these weren''t hacks but valid changes you could make either by using custom files or modifying console variables. Would you call that cheating? I would.

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The camera position trick has been disabled in most online gaming servers for Quake 3, because that can legimately be used for cheating.

As for FOV, you cannot see more around you by changing the command cg_fov in Quake 3. It simply distorts the perception of everything.

You can use it for zooming, but zooming shouldn''t be considered cheating either.

I think cheating is using any 3rd party assist to give you abilities that normal players would not possess by simply playing the game. Changing the brightness to make things clearer is not cheating. It''s an option you''re allowed to tinker with. Anyone else can do it too.

Just because a newbie to Quake 3 wouldn''t know to do things like lower texture quality, does not mean that it''s cheating. Eventually, they will most likely discover options like this. That''s part of the fun.

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OctDev--
I would probably half and half it. Since you must have a server, I''d leave a yes/no option for a lot of things. But this is an independant, like q3. If it was a subscription mmorpg, then I would go hard and not allow much in the way of gameplay customization, and for the very reasons given, unfair advantage. Not that everything would follow set paths, I would allow customization thats worked for in my set design. But things like keybinding scripts, not a chance. My game would test your ability to roleplay it based on rules, not geek it apart with your knowledge of being a programmer at id software.

-> Will Bubel
-> Machine wash cold, tumble dry.

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Games generate a creative space; people use that creative space in a way that pleases them.

Is it the player''s fault for being selfish? Is the game''s fault for not reacting to the player''s choices?

Are you going to introduce morality? How do you introduce morality in a game when morality begins with the player?

Should the issue really be "the fruits of gaming"?

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Bind all ya want in Quake, I''ll still kick yer ass.

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maybe games should have less graphical intense enviroments that did not require graphics tweaking on lower end hardware for decent (50+) framerates? maybe levels should be designed to be lit properly using the entire spectrum of colors and not have pitch black areas surrounded by very dark areas. some moniters are darker then others. brightness changes are normal. if the area is close to black increasing the brightness merely washes the video put causing some glare. gamma controls will cause an increase in contrast since now darker ares get lighter, and is actually designed to fix moniters. NO moniter handles color linearly, so gamma correction fixes this by apply the reletive color brightness in a non linear fashion. thus the screen gets darker or lighter quicker.

ALL those options are easily found by anyone who clicks options. while i agree to an extent that veterns have an advantage over nwbies when they sue scripts to help perform complex manuvers, by the same token ANYONE who decides it worth their time and effort to create such scripts should use them. its part of the game. its part of the game, no one has to download anything special or requires anything beyond the game. sure some obscure commands require the console command reference, but someone who wants the knowledge can get it easily (if not then they are not playing online).

why punish the vetern so the newbie can play better? CS has done this (cant move hostages, less accurate weapons, etc) and the gameplay has gone down hill.

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So...

3rd party additions are cheating. It is additional code, others don''t have it, it is an unfair advantage.

Client side variables are still a grey area, by consensus. However, if the developer put them there to be adjusted, I lean towards any adjustment being ok.

As far as public vs. subscription servers, I agree. If the servers are being run completely by the developing company for some sort of consisteny (be that subscription based play, etc), then consistency must be uphold by limiting customization. For privately held but publicly available servers, give the servers limiting options, with several generic ''gameplay option'' templates.

There is no way to make a player moral; in fact, many play games just to be immoral. So it lies upon the developer to force any morality that they deem necessary for gameplay. This ties in nicely with the server situations above, and leads to the next observation.

If a game must force morality then it must attempt to level the playing environment. Cutting edge hardware requirements are a poor design choice for games forcing morality (and disabling most customization) on their players.

Finally, I think the best point is a person''s final one--why punish the veteran so the newbie can play better? Always keep that in mind, as your veterans are likely to continue generating \$ and word of mouth publicity.

--OctDev

The Tyr project is coming...

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I wouldn''t say scripting is cheating so much. What I''m saying is an environment that lacks rules almost always has thoses that take full advantage of the lack of control. I would rather see a detailed ruleset with toggle options for a q3 type game.

To throw a question at you, is it cheating to use item glitches in rpgs to get yourself a full inventory of a certain type of item? My answer would be yes, because its not something that the developers set aside when they wrote the rules.
How about an enemy in the game that gives of bazillions of exp, and has 1 hp and 0 in all of it''s attributes? Not cheating. It''s a clear and intentional addition to the game.
Scripts in an online shooter? Clear and intentional feature addition to the engine, I say its not cheating.

-> Will Bubel
-> Machine wash cold, tumble dry.

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Perhaps I was unclear. I think we agree. Scripts are not really cheating, as they are added by the developer for the gamer. I think that it would be ludicrous for developers to include scripting engines for gamers in certain types of games (in a MMORPG, for instance, rules need to be established), however, it works very well in FPSs.

I would say that if you are taking advantage of a situation in a game that is later patched by the developrs, it is a clear cut indication that you were cheating by manipulating an unintended side effect of the code. If it is a feature that is intentionally provided, then you are free to pass go.

It really just comes down to a design choice of what to include. When should environments become overly customizable? We''ve already covered some of this. Basically, when it provides an unfair advantage. Is scripting by veterans an unfair advantage in a FPS? No. In much the same way letting someone have an 80 level character and a newbie starting with a 1 level character in a RPG. Experience creates both veteran situations. Yes, the experience is not handled strictly the same way, but it comes down to experience in both situations.

--OctDev

The Tyr project is coming...

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# 12.3 The regression equation  (Page 4/8)

 Page 4 / 8

The correlation coefficient, r , developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y .

The correlation coefficient is calculated as

$r=\frac{n\Sigma \left(xy\right)-\left(\Sigma x\right)\left(\Sigma y\right)}{\sqrt{\left[n\Sigma {x}^{2}-{\left(\Sigma x\right)}^{2}\right]\left[n\Sigma {y}^{2}-{\left(\Sigma y\right)}^{2}\right]}}$

where n = the number of data points.

If you suspect a linear relationship between x and y , then r can measure how strong the linear relationship is.

## What the value of r Tells us:

• The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
• The size of the correlation r indicates the strength of the linear relationship between x and y . Values of r close to –1 or to +1 indicate a stronger linear relationship between x and y .
• If r = 0 there is absolutely no linear relationship between x and y (no linear correlation) .
• If r = 1, there is perfect positive correlation. If r = –1, there is perfect negativecorrelation. In both these cases, all of the original data points lie on a straight line. Of course,in the real world, this will not generally happen.

## What the sign of r Tells us

• A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation) .
• A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation) .
• The sign of r is the same as the sign of the slope, b , of the best-fit line.

## Note

Strong correlation does not suggest that x causes y or y causes x . We say "correlation does not imply causation."

The formula for r looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate r . The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).

## The coefficient of determination

The variable r 2 is called the coefficient of determination and is the square of the correlation coefficient, but is usually stated as a percent, rather than in decimal form. It has an interpretation in the context of the data:

• ${r}^{2}$ , when expressed as a percent, represents the percent of variation in the dependent (predicted) variable y that can be explained by variation in the independent (explanatory) variable x using the regression (best-fit) line.
• 1 – ${r}^{2}$ , when expressed as a percentage, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.

Consider the third exam/final exam example introduced in the previous section

• The line of best fit is: ŷ = –173.51 + 4.83x
• The correlation coefficient is r = 0.6631
• The coefficient of determination is r 2 = 0.6631 2 = 0.4397
• Interpretation of r 2 in the context of this example:
• Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained by the variation in the grades on the third exam, using the best-fit regression line.
• Therefore, approximately 56% of the variation (1 – 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best-fit regression line. (This is seen as the scattering of the points about the line.)

## Chapter review

A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called “errors,” measure the distance from the actual value of y and the estimated value of y . The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data.

The correlation coefficient r measures the strength of the linear association between x and y . The variable r has to be between –1 and +1. When r is positive, the x and y will tend to increase and decrease together. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. The coefficient of determination r 2 , is equal to the square of the correlation coefficient. When expressed as a percent, r 2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line.

Use the following information to answer the next five exercises . A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10

Draw a scatter plot of the data.

Use regression to find the equation for the line of best fit.

ŷ = 2.23 + 1.99 x

Draw the line of best fit on the scatter plot.

What is the slope of the line of best fit? What does it represent?

The slope is 1.99 ( b = 1.99). It means that for every endorsement deal a professional player gets, he gets an average of another \$1.99 million in pay each year.

What is the y -intercept of the line of best fit? What does it represent?

What does an r value of zero mean?

It means that there is no correlation between the data sets.

When n = 2 and r = 1, are the data significant? Explain.

When n = 100 and r = -0.89, is there a significant correlation? Explain.

Yes, there are enough data points and the value of r is strong enough to show that there is a strong negative correlation between the data sets.

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***keydifferences.com/difference-between-sample-mean-and-population-mean.html
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Terry
what is statistic?
it's a science of collection, organization, analysis and summarizing data to get useful information to make several types of conclusions.which can be used in real life.
anshika
what is the statistical probability that president Trump will remain in the white house after the election of 2020?
Terry
i agree with anshika is right but let me add that such decisions are made in face of uncertainty
Maureen
yes
Stephen

format: plain


text:
## Experimental Mathematics

### New representations for the Madelung constant

Richard E. Crandall

#### Abstract

From a modern theta-function identity of G. E. Andrews we derive new representations for the celebrated Madelung constant and various of its analytic relatives. The method leads to connections with the modern theory of multiple zeta sums, generates an apparently entire "$\eta$ series" representation, and, for the Madelung constant in particular, yields a finite-integral representation. These analyses suggest variants of the Andrews identity, leading in turn to number-theoretical results concerning sums of three squares.

#### Article information

Source
Experiment. Math. Volume 8, Issue 4 (1999), 367-379.

Dates
First available in Project Euclid: 9 March 2003


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# What is the average speed of an object that is still at t=0 and accelerates at a rate of a(t) =t^2-4 from t in [2, 3]?

Jan 12, 2018

4.58 $\frac{m}{\sec}$

#### Explanation:

Acceleration of the object is given as,$a = 4 {t}^{2} - 4$
So,velocity will be $\frac{{t}^{3}}{3} - 4 t$ (by integrating)
Hence,displacement will be,
$S = \frac{{t}^{4}}{12} - \left(2 {t}^{2}\right)$ (by integrating)
Hence,displacement with in t=2 to t=3 will be -4.58 meter
Now here distance=displacement that has happened in between 2 sec and 3 sec
Hence average speed with in this duration=(total distance covered/total time)
I.e $\left(\frac{4.58}{1}\right)$ $\frac{m}{\sec}$ or 4.58 $\frac{m}{\sec}$


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Hi i''m writing a little conosle application with C++, which is virutal pet "tamagotchi, and i;m stuck with a problem. I need to know how to write the command where after each hour it adds a certain amount of value to a varible. Is this method called ticks or something, can someone guide me on how to do this?? Thanks Kev

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If you do this by ticks, you have to call timeGetTime(), this function return the miliseconds since the machine started, you have to save a time_start variable and in your update, check the miliseconds that have passed since time_start.

Hope this help

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Sitio de desarrollo de videojuegos en español
Digital Moon Studio...

[edited by - NeonGE on July 2, 2003 1:46:21 PM]

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What i''m trying to do is for every hour or so it adds 1 to the hunger varible. can any one show me how to write the statement or syntax for it please, ive been struggling with this.

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timebeginperiod(1);int start_time=timegettime();int current_time=start_time;int hunger=0;while(true){current_time=(timegettime()-start_time);hunger+=current_time/3600000;}timeendperiod(1);

Note this is just one (probably clumsy) way to do it. The "magic number" of 3600000 is the number of milliseconds in an hour since timegettime returns milliseconds.

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## Bayes Theorem: Certainty Starts in Here

Continuing on my series on Bayes’ Theorem, recall that the question of any rational investigation is this: what is the probability of the theory of interest T, given everything that I know K? Thanks to Bayes’ theorem, we can take this probability $p(T | K)$ and break it into manageable pieces. In particular, we can divide K into background information B and data D. Remember that this is just convenience, and in particular that B and D are both assumed to be known.

Suppose one calculates $p(T | DB)$ for some theory, data and background information. Think of it as a practice problem in a textbook. This calculation, in and of itself, knows nothing of the real world. So what follows? We can think of the probability as a conditional if-then statement:

1. If DB, then the probability of T is $p(T | DB)$.

To draw a conclusion from this, we must add the premise.

2. DB.

Only then can we conclude,

3. The probability of T is $p(T | DB)$.

But wait a minute … the whole point of this exercise was to reason in the face of uncertainty. Where do we get the nerve to simply assert 2, that DB is true? Where is the inevitable uncertainty of measurement? Isn’t treating the data as certain hopelessly idealized? Shouldn’t we take into account how probable DB is? But there are no raw probabilities, so with respect to what should we calculate the probability of DB? We’re headed for an infinite regress if we keep asking for probabilities. How do we get premise 2? Are probabilities all merely hypothetical?

We can put this another way: what counts as data, given that it must be treated as certain? I can know some necessary truths, like mathematical and logical truths, with something like certainty, but what about the real world? It is often said, in a kind of “of course” tone of voice, that all observation is theory laden. If that is the case, then there would be nothing to be taken as given, and all probabilities would be merely hypothetical. Science would just be a game of “imagine if …”. To overcome this, there must be some raw data from which we can begin.

Perhaps you think that there are some ideas that are practically certain, that we can just assume those and get on with it. But which ones? Remember that these are the statements from which we calculate probabilities, so we can’t just say that any statement with a probability greater than 0.999 … is practically certain. In any case, I think we can do better.

Let me head out on a limb (I still haven’t totally convinced myself). I think that part of the answer is the certainty of first-person experience. There seem to be contingent truths of which I can be certain. I can be certain that I am having the experience of seeing a red chair. There may not actually be a red chair – I may be dreaming, the light may be fooling me, I may be colour-blind, and so on. But I know my own thoughts immediately and incorrigibly. I can be certain that I think that there is a red chair. I know my own mind immediately, directly, without having to infer it from anything else.

We now at least have a starting point. If I want to know whether there really is a red chair over there, I will need to calculate the probability of there being a red chair over there, given that I think there is a red chair, plus the sum total of my other experiences of the world and any other necessary truths I know. That probability may be not be obvious, or may turn out to be lower than I expect, but at least we are asking more than a hypothetical question.

These rather abstract considerations have a practical application: the data D (and the background B, for that matter) should be as raw as possible. Jaynes gives the example of an experiment in which a psychic predicts a randomly drawn card with a success rate much higher than chance. What is the data? If we take D = the experiment, then we are liable to be too easily convinced of psychic powers. But D is not given. What I know is that that some ESP experimenter claimed that the experiment occurred as described. Psychic powers remain a possible explanation, but now several others come into view – hoax, poor procedure, gullibility, selective reporting etc.

In practice, we rarely need to retreat all the way back to subjective experience to draw useful probability conclusions. But they at least provide a bedrock of contingent yet certain beliefs from which we can build. The regress stops here.


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# How to calculate concentrations in a diluted mixture of multiple solutions?

I have a homework problem as follows:

A solution was prepared by mixing $25.0\ \mathrm{mL}$ of $0.208\ \mathrm{M}$ $\ce{Fe(NO3)3}$ and $4.50\ \mathrm{mL}$ of $0.00190\ \mathrm{M}$ $\ce{KSCN}$ and diluting to $50.00\ \mathrm{mL}$ with solvent, $0.10\ \mathrm{M}$ $\ce{HNO3}$. Calculate the initial concentrations of the thiocyanate ion and the iron(III) ion before any reaction takes place.

Now, how do I proceed with the question? At first I thought that it should just be the same concentration as the solutions, $0.208\ \mathrm{M}$ and $0.00190\ \mathrm{M}$, because there is only one thiocyanate and one iron(III) ion in the solutions. But the fact that we have an $\ce{HNO3}$ solution confuses me. Am I somehow supposed to use $M_1 V_1 = M_2 V_2$?

• Welcome to Chemistry.SE. Take the tour to get familiar with this site. This appears to be a homework question, please share your thoughts and attempts towards the solution. It'll make us certain that we aren't doing your homework for you.
– user15489
Aug 29, 2015 at 23:38
• Nitric acid is like the question said just the solvent, so you are on the right track with your formula, as it states before any reaction. Aug 30, 2015 at 10:50

Am I somehow supposed to use $$M_1V_1=M_2V_2$$?

Yep!

The problem has given you three different solutions:

1. Iron(III) nitrate
2. Potassium thiocyanate
3. Nitric acid

You know the concentrations of each, and you know (or can calculate) the volumes of each that are mixed to yield the final solution. What you'll actually be using in the calculation are three different equations:

1. $$M_{1i}V_1 = M_{1f}V_f$$
2. $$M_{2i}V_2 = M_{2f}V_f$$
3. $$M_{3i}V_3 = M_{3f}V_f$$

$$M_{\#i}$$ is the initial concentration of each of the three chemicals in their separate starting solutions, and $$M_{\#f}$$ is the final concentration of each after mixing, that the problem is asking for (except for the $$\ce{HNO3}$$).


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Graphing Two-variable Inequalities, Sudden Impact Biv 10, Nobody's Fool Full Movie, James Karen Age, Sbcc Parking Permit Motorcycle, Google Forms Shopping List, Bending Square Tubing By Hand, Hyundai Elantra Radio Fuse, " />

# what is block diagram in control system

## what is block diagram in control system

In other words, practical representation of a control system is its block diagram. We will vary this input with some mechanism. Consider a simple R-L circuit Apply KVL Now taking laplace transform of Eq.1 and Eq.2 with initial condition zero From eq.3 and eq.4 From fig: Now taking laplace transform of Eq.5, and Eq.6 For the right-hand side of eq.5, we will use a summing point. Signal Factor (inputs) pass through the design of the product and is output into measured Response Variable (also called the Ideal Function). Find the transfer function $$\frac{X(s)}{F(s)}$$ for the block diagram shown in the figure. If we define the output of the first system as h(t), we can define h(t)as: 1. h ( t ) = x ( t ) ∗ f ( t ) {\displaystyle h(t)=x(t)… What is a SysML Block Definition diagram? Find the transfer functions for the following block diagrams. Calculate the natural frequency, ω n , and damping ratio, ζ, for the plant. Next Page . A control system is a system, which provides the desired response by controlling the output. An important factor to remember is that the block diagram represents flowpaths of control signals, but does not represent flow of energy through the system or process. It is easier and better to derive the transfer function of the control element connected to the system, separately. Control Systems - Block Diagram Reduction. Here, resultant input signal is the summation of all input signals applied. Representation of a common input signal to more than one blocks of a control system is done by a common point as shown in the figure below with point X. Note that the input does not get divided at a point, but instead the input propagates through all the paths connected to that point without affecting its value. From the geology perspective, it is a perspective representation of the geology of an area showing surface contours and generally including two vertical cross sections. Click here for a blank P-Diagram form in MS Excel. The resultant signal is the input of a control system block of transfer function G(s) and the final output of the system is Feedback Control System Block Diagram. What is Block Diagram A bock diagram is pictorial representation of the functions … For (b) and (c), also calculate the damping ratio, ζ, and natural frequency, ω n . There are two input signals R(s) and ± X(s) entering in a summing point at position A. A block diagram is especially focused on the input and output of a system. • To be able to select controller parameters to m eet design objectives. We are a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for us to earn fees by linking to Amazon.com and affiliated sites. For simplifying a complex control system, block diagrams are used. When a train enters a block, signals at both ends change to indicate that the block is occupied, typically using red lamps or indicator flags. If you’re looking to do some control systems study, check out our control systems MCQs. The signal R(s) at take off point A will become G(s)R(s) at point B. A block diagram is a visualization of the control system which uses blocks to represent the transfer function, and arrows which represent the various input and output signals. The block system is referred to in the UK as the method of working, in the US as the method of operation, and in Australia as safeworking. Hence, by having a take off point the same input signals can be applied to more than one systems or blocks. If the integral gain constant, K I , is set to zero, calculate the value of K P required for the natural frequency of the closed loop system to be 6 rad/s. • To be able to represent a control system with block diagrams. This algebra deals with the pictorial representation of algebraic equations. Not affiliated How to Make a Block Diagram. Here R(s), X(s) and Y(s) are the input signals. In addition to that, the diagram also shows there is a feedback path through which output signal C(s) is fed back and compared with the input R(s) and the difference between input and output is , which is acting as the actuating signal or error signal. Block diagrams derive their name from the rectangular elements found in this type of diagram. Electrical4U is dedicated to the teaching and sharing of all things related to electrical and electronics engineering. PLC replaced the analog control system by bringing the digital control system … Ein Blockdiagramm (auch Blockbild oder Blockschema) ist ein meist 2-dimensionales, grafisches Diagramm eines mechanisch-elektrischen Systems oder eines Berechnungsschemas, mit dessen Hilfe die Funktion des Systems beschrieben werden kann. Cite as. In other words – if there are more than one summing points directly inter associated, and then they can be easily interchanged from their position without affecting the final output of the summing system. Where C(s) is the output and R(s) is the input of that particular block. Calculate the transfer function $$\frac{X(s)}{F(s)}$$ in terms of the feedback parameter K. For the case where K = 0 and the input, f(t), is a unit delta function, δ(t), calculate x(t) using Laplace transforms. Lecture 5 for Control Systems Engineering (UFMEUY-20-3) and Industrial Control (UFMF6W-20-2) at UWE Bristol. But overall transfer function of the system is the ratio of transfer function of final output to transfer function of initial input of the system. You will understand the working principle of PLC from the block diagram of PLC. Repeat part (d) when the integral gain constant, K I , is equal to 1. • Transfer functions, block diagrams and simplification • Feedback controllers • Control system design output input----- = fD() The general form x F---4 + D D2 ++4D 16 = -----An example. Advertisements. The open-loop control system block diagram is shown below. Block Diagram Reduction Rules. Since, the output is controlled by varying input, the control system got this name. In a closed loop control system, a fraction of output is fed-back and added to input of the system. is a constant feedback parameter. If the proportional gain constant, K P , is equal to 20 and the commanded position is a step input x set  = 3 ⋅ u(t), calculate the steady state error where the integral gain constant, K I , is equal to zero.


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# Thermonuclear fusion in the Sun

## Homework Statement

Our present theory of the sun and the mechanism of thermonuclear fusion inside it predicts that the following reactions occur in the sun.
1H + 1H --> 2H + antielectron + neutrino
1H + 1H --> 2H + antielectron + neutrino
electron + antielectron --> photon + photon
electron + antielectron --> photon + photon
2H + 1H --> 3He + photon
2H + 1H --> 3He + photon
3He + 3He --> 4He + 1H+ 1H

A problem arises in this theory due to one of its calculated count of something not matching with the observational counting. So what is that ‘something’ and what is the relation between the prediction and observation (i.e. greater or smaller)?

## Homework Equations

The overall fusion reaction
4 1H + 2 e --> 4He + 2 neutrinos + 6 photons


text:
# Homework Help: Doppler Shift: Communication from a Space Probe

1. Mar 21, 2014

### blue_lilly

1. The problem statement, all variables and given/known data
Space probes communicate with controllers on Earth using radio waves. Suppose a probe has a radio-transmitter that operates at a frequency of 516.0 MHz when at rest relative to the observer. This probe is now wandering the solar system. The signal you pick up from the probe is shifted 5.955E+4 Hz higher in frequency than the rest frequency given above. How fast is the probe traveling and is it traveling towards or away from Earth? (assume it is either traveling directly towards or directly away from Earth. You must get both answers correct at the same time.)

2. Relevant equations
c=3E8
ƒobserved = ƒsourse (1 $\pm$ (rel Velocity/c))

3. The attempt at a solution
I know that this is a one way Doppler Shift, I am the observer and the source is the probe. Because it a Doppler Shift I am going to use this formula, ƒobserved=ƒsourse(1$\pm$(rel V/c))

ƒo = ƒs(1$\pm$(rel V/c))
ƒs: I am given the ƒs in MHz so I need to convert to Hz.
516MHz = 516000000Hz = 5.16E8Hz
ƒo: In the problem, it says "The signal you pick up from the probe is shifted 5.955E+4Hz (59550Hz) higher in frequency". This means that the ƒrequency that is obsorbed is the ƒrequency of the source plus the amount it is shifted.
ƒo = ƒs + shift
ƒo = 516000000Hz + 59550Hz = 516059550Hz = 5.1605955E8Hz
c: The speed of light is 3E8.
relV: We are solving for this.

ƒo = ƒs(1$\pm$(relV /c))
516059550Hz = 516000000Hz (1-( relV / 3E8 ))
59550Hz = 1-( relV / 3E8 )
59549Hz = relV / 3E8
(59549Hz)(3E8) = relV
relV = 1.78647E13 m/s

So the relV is 1.78647E13 m/s and the direction would be towards earth because the ƒo is louder then the ƒs.

But the answer is incorrect and I am not sure what I did wrong.
Any help would be GREATLY appreciated.

2. Mar 21, 2014

### paisiello2

How can anything travel faster than light? 1.78647E13 m/s is a huge number.

I think you simply did an arithmetic mistake in step 3 of solving for relV. You would have caught this had you put in all the units.

Lastly, a frequency isn't "louder" than another frequency. It is simply higher or lower.

3. Mar 21, 2014

### Staff: Mentor

Check the algebra in this step:

As paisiello2 says, you should carry units along in your calculations, to help you catch mistakes like this...

4. Mar 21, 2014

### blue_lilly

I keep trying to fix the carry over numbers but its not working very well, so sorry if it looks really confusing.

ƒo = ƒs(1±(relV /c))
516059550Hz = 516000000Hz (1-( relV / 3E8 ))
-516000000Hz -516000000Hz
59550Hz = 1-( relV / 3E8 )
-1 -1 [subtract 1 from both sides]
59549Hz = - (relV / 3E8)
x(3E8) x(3E8) [multiply both sides by 3E8]
(59549Hz)(3E8) = -relV
-relV = (59549Hz)(3E8)
relV = -1.78647E13 m/s [reverse neg sign]

I rewrote the problem in my note book and the answer is still wrong. Am I not seeing the algebra mistake?

5. Mar 21, 2014

### Staff: Mentor

The algebra mistake is still here:

The units error is that you have Hz on the LHS of the equation, and the RHS is unitless.

That's because you subtracted 516000000Hz from both sides, instead of...

6. Mar 21, 2014

### blue_lilly

Does this fix the unit error? 516059550Hz = 516000000Hz (1-( relV / 3E8m/s^2 ))

Instead of subtracting 516000000Hz, should I have added 516000000Hz to both sides?

7. Mar 21, 2014

### Staff: Mentor

Nope.

If you want to solve this equation: 15x = 30

What do you do to isolate x? You certainly don't subtract 15 from both sides...

8. Mar 21, 2014

### blue_lilly

ohhh, i feel like an idiot

ƒo = ƒs(1±(relV /c))
516059550Hz = 516000000Hz (1-( relV / 3E8 ))
/516000000Hz /516000000Hz
1.000115407Hz = 1-( relV / 3E8m/s^2 )
1.154067E-4 = - relV / 3E8m/s^2
x(3E8m/s^2) x(3E8m/s^2)
(1.154067E-4)(3E8m/s^2)
34622.09 = - relV
- relV = 34622.09
relV = - 34622.09 m/s^2

SO does this look right then? Also would the neg sign mean that it is traveling away from you?

9. Mar 21, 2014

### Staff: Mentor

Closer. The units are not correct, though. See my edits above...

The units of velocity are m/s, not m/s^2. And when you divided Hz/Hz in the first step, that makes the LHS dimensionless, just like the RHS. The units of the LHS and RHS of any equation have to match.


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# Year 12 Regular Problem Solving Classes

Middlesbrough

## Overview

A series of maths problem-solving sessions designed to give students the confidence to apply for university courses that require or take into consideration achievement in an admissions test.

These regular classes offer students the opportunity to develop mathematical problem-solving skills through discussion and collaboration. They are designed to help students to develop strategies and confidence when tackling unfamiliar problems in maths and will help with preparations for taking advanced papers such as the MAT, TMUA and STEP examinations.

At the same time, the problems used in the course are fun and rewarding. Attending the sessions will greatly enrich students’ mathematical experience and help them to develop a better understanding of A level Mathematics.

## Aims

• To develop initial strategies when dealing with maths problems
• To develop confidence when dealing with maths problems
• To provide some initial information about the problem solving involved in university admissions tests
• To provide a platform on which to build secure problem solving techniques
• To develop tenacity when dealing with maths problems

## Who will benefit from attending?

The course is designed for any A level Mathematics students who have an enquiring mind and wish to develop their problem solving ability for their A level studies and beyond.

It is particularly useful for those students who wish to make the first steps in preparing for university admissions tests such as the MAT, TMUA and STEP examinations.

## Content

The course covers a wide range of mathematical disciplines with problems. These can include

• Algebra: the difference between two squares and other identities
• Geometry: angles, triangle and circles
• Number: digits and divisibility
• Algebra: forming and solving equations
• Combinatorics: systematic counting
• Number: prime factorisation, fractions and irrationals
• Algebra: sequences and series
• Number: indices and logarithms
• Algebra: quadratics, cubics and other polynomials
• Geometry: trigonometry
• Combinatorics: further systematic counting and placement
• Geometry: coordinates and vectors
• Calculus: curve sketching and differentiation
• Calculus: integration
• Combinatorics: the binomial expansion

## Materials and Equipment

Students do not need any special equpiment for the course although a smartphone with the GeoGebra or the Desmos apps installed may be useful.

## Other Information

The following problems provide a taste of the sort of problem solving that will be encountered in the classes

Problem 1: How many primes greater than two can be found that are one less than a square number?

Problem 2: How many pairs of integers can you find that satisfy the equation $$x^2–y^2=45$$?

## Study Schedule

Last session is this year's series which is going to take place on the 13th July is welcoming year 11 students who will have a chance to solve problems relevant in higher mathematics, so we reccommend it to students who will continue Maths post 16.

Each 4 hour session is essentially stand-alone so there is no requirement necessarily to attend all five sessions, although we would encourage students to attend as many sessions as they can. Students who have attended similar sessions in previous years can expect to meet plenty of new materials this year. Year 12 and 13 students will benefit most but Year 10 and 11 who plan to study Maths post 16 are also welcome.

Sessions will take place, 9:30-13:30 on the following dates:

• 24 November 2018
• 2 February 2019
• 16 March 2019
• 4 May 2019
• 13 July 2019

We aim to start by 9:30 am and finish by 1:30pm, with some short breaks in between. We cannot provide refreshments but there are wending machines on the site or students can bring food and drinks along with them.
We advise students to bring their own pens, pencils, paper and other equipment. If they wish, they may also bring calculators or other technological devices. We will welcome students at the Main Reception.

Middlesbrough Colllege campus and is a 2 min walk from the train station and 5 min walk from the MIddlesbrough bus station.
If you are coming by car, the best thing would be to park on the side road for £ 1.20 as the parking is not free.

https://www.northernskills.co.uk/contact-us/middlesbrough-college-campus

Please email [email protected] for any questions related to this course.

## Key Facts

#5323

Students

Year 12

##### Curriculum focus:

University admissions tests, A level Mathematics, A level Further Mathematics

##### Mathematical focus:

Pure, Problem solving

Student course

20 hours

North East

##### Venue:

Middlesbrough College, Sixth Form building (MC6),
Dock Street,
Middlesbrough,

##### Date:

Sat 13th Jul 2019

##### Course times:

09:30 - 13:30

Printable Version


text:
# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 5.3.4 Scaffolds of Cocartesian Fibrations

Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty$-categories indexed by $\operatorname{\mathcal{C}}$. Our goal in this section is to show that the diagram $\mathscr {F}$ can be recovered, up to equivalence, from the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Definition 5.3.3.1. More precisely, we will show that there exists a levelwise categorical from $\mathscr {F}$ to the strict transport representation $\operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$ of Construction 5.3.1.5 (Corollary 5.3.4.19).

We begin with some general remarks. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be any cocartesian fibration of simplicial sets and let $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the weak transport representation of $U$ (Construction 5.3.1.1). Every levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ can be viewed as a natural transformation from $\mathscr {F}$ to the weak transport representation $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, which we can identify (using Corollary 5.3.2.23) with a morphism from the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ into $\operatorname{\mathcal{E}}$. Our first goal is to give an explicit characterization of the collection of morphisms $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which arise in this way, which we will refer to as scaffolds of the cocartesian fibration $U$ (Definition 5.3.4.2 and Remark 5.3.4.10).

Definition 5.3.4.1. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $e$ be an edge of the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$. Let us identify $e$ with a pair $(f, \overline{e} )$, where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $\overline{e}$ is an edge of the simplicial set $\mathscr {F}(C)$. We will say that the edge $e = (f, \overline{e} )$ is horizontal if $\overline{e}$ is a degenerate edge of $\mathscr {F}(C)$.

Definition 5.3.4.2. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. We will say that a morphism of simplicial sets $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a scaffold if it satisfies the following conditions:

$(0)$

The diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [rr]^{\lambda } \ar [dr] & & \operatorname{\mathcal{E}}\ar [dl]_{U} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & }$

is commutative (where the left vertical map is the projection map of Construction 5.3.2.1).

$(1)$

The morphism $\lambda$ carries horizontal edges of $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the induced map

$\mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow {\lambda } \{ C \} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$

is a categorical equivalence of simplicial sets.

Example 5.3.4.3. Let $n$ be a nonnegative integer and let $\operatorname{\mathcal{E}}$ denote the nerve of the partially ordered set $Q = \{ (i,j) \in [n] \times [n]: j \leq i \}$. Then there is a cocartesian fibration of $\infty$-categories $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$, given on vertices by the formula $U(i,j) = i$. Let $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by $\mathscr {F}(i) = \Delta ^ i$, so that vertices of the homotopy colimit can be identified with elements of $Q$. There is a unique morphism of simplicial sets $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which is the identity at the level of vertices, which is a scaffold of the cocartesian fibration $U$. Moreover, $\lambda$ is monomorphism, and an $n$-simplex $(i_0, j_0) \leq (i_1, j_1) \leq \cdots \leq (i_ n, j_ n)$ belongs to the image of $\lambda$ if and only if $j_ n \leq i_0$. The case $n=3$ is depicted in the following diagram, where the image of $\lambda$ is indicated with solid arrows:

$\xymatrix@C =30pt@R=30pt{ & & & (3,3) \\ & & (2,2) \ar@ {-->}[ur] \ar [r] & (3,2) \ar [u] \\ & (1,1) \ar@ {-->}[ur] \ar [r] & (2,1) \ar [u] \ar [r] & (3,1) \ar [u] \\ (0,0) \ar@ {-->}[ur] \ar [r] & (1,0) \ar [r] \ar [u] & (2,0) \ar [u] \ar [r] & (3,0). \ar [u] }$

Example 5.3.4.4. Let $\operatorname{\mathcal{E}}$ be an $\infty$-category equipped with cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ having fibers $\operatorname{\mathcal{E}}_0 = \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}_{1} = \{ 1\} \times _{ \Delta ^1} \operatorname{\mathcal{E}}$. Choose a functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ and a morphism $h: \Delta ^1 \times \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}$ which witnesses $F$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$, in the sense of Definition 5.2.2.4. Then $F$ can be identified with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{QCat}$, and the map

$\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) = ( \Delta ^1 \times \operatorname{\mathcal{E}}_0 ) \coprod _{ ( \{ 1\} \times \operatorname{\mathcal{E}}_0)} \operatorname{\mathcal{E}}_1 \xrightarrow {(h,\operatorname{id})} \operatorname{\mathcal{E}}$

is a scaffold.

Remark 5.3.4.5 (Isomorphism Invariance). In the situation of Definition 5.3.4.2, suppose that we are given a pair of morphisms $\lambda , \lambda ': \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ which are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}})$. Then $\lambda$ is a scaffold if and only if $\lambda '$ is a scaffold (see Corollary 5.1.2.5 and Remark 4.5.1.15).

Remark 5.3.4.6 (Change of $\operatorname{\mathcal{E}}$). Suppose we are given a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{T} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{ U' } \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), & }$

where the vertical maps are cocartesian fibrations and $T$ is an equivalence of cocartesian fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then a morphism $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a scaffold of the cocartesian fibration $U$ if and only if $T \circ \lambda$ is a scaffold of the cocartesian fibration $U'$.

We now describe two important examples of scaffolds, both of which can be regarded as generalizations of Example 5.3.4.3.

Construction 5.3.4.7 (The Universal Scaffold). Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\operatorname{sTr}_{ \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denote the strict transport representation of $U$ (Construction 5.3.1.5). For each $n \geq 0$, we can identify $n$-simplices of the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$ with pairs $(\sigma , \tau )$, where $\sigma$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (given by a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$) and $\tau$ is an $n$-simplex of the $\infty$-category $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C_0) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0 / } ), \operatorname{\mathcal{E}})$, which we identify with a morphism of simplicial sets $\Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ C_0/ }) \rightarrow \operatorname{\mathcal{E}}$. Let us identify the diagram $C_0 \xrightarrow {\operatorname{id}} C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ with an $n$-simplex $\widetilde{\sigma }$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0/} )$, and let $\lambda _{u}( \sigma , \tau )$ denote the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map

$\Delta ^ n \xrightarrow { (\operatorname{id}, \widetilde{\sigma }) } \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C_0/} ) \xrightarrow {\tau } \operatorname{\mathcal{E}}.$

The construction $(\sigma , \tau ) \mapsto \lambda _{u}( \sigma , \tau )$ determines a morphism of simplicial sets

$\lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{E}},$

which we will refer to as the universal scaffold of the cocartesian fibration $U$.

Proposition 5.3.4.8. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories. Then the morphism $\lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{E}}$ of Construction 5.3.4.7 is a scaffold, in the sense of Definition 5.3.4.2.

Proof. It is clear that the composition $U \circ \lambda _{u}$ coincides with the projection map $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $e$ be a horizontal edge of the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$, determined by a morphism $\overline{e}: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$ together with a degenerate edge $\operatorname{id}_{T}$ of the simplicial set $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( C)$. Identifying $T$ with an object of the $\infty$-category $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C / } ), \operatorname{\mathcal{E}})$, we see that $\lambda _{u}(e)$ coincides with the morphism $T( \overline{e} )$ and is therefore a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$. To complete the proof, we observe that for every object $C \in \operatorname{\mathcal{C}}$, the induced map

$\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \xrightarrow {\lambda } \{ C \} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$

agrees with the map $\operatorname{ev}_{C}: \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C / } ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ given by evaluation on the initial object $\operatorname{id}_{C} \in \operatorname{\mathcal{C}}_{C/}$, and is therefore a trivial Kan fibration of simplicial sets (Proposition 5.3.1.7). $\square$

Corollary 5.3.4.9. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories. Then there exists a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$.

Remark 5.3.4.10 (Universality). Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Applying Corollary 5.3.2.23, we obtain a bijection from the set of morphisms $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ in the category $(\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the set of natural transformations $\alpha : \mathscr {F} \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Unwinding the definitions, we see that $\alpha$ factors through the subfunctor $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \subseteq \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if $\lambda$ satisfies condition $(1)$ of Definition 5.3.4.2. If this condition is satisfied, then $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is a levelwise categorical equivalence if and only if $\lambda$ satisfies condition $(2)$ of Definition 5.3.4.2. We therefore obtain a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Levelwise categorical equivalences \alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}} \} \ar [d]^{\Phi } \\ \{ \textnormal{Scaffolds \lambda : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}} \} . }$

Concretely, this bijection carries a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ to the composite map

$\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow { \alpha } \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \xrightarrow { \lambda _ u } \operatorname{\mathcal{E}},$

where $\lambda _{u}$ is the universal scaffold of Construction 5.3.4.7.

Construction 5.3.4.11 (The Taut Scaffold). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Suppose we are given an $n$-simplex of the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is a pair $(\sigma , \tau )$, where $\sigma$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (given by a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$) and $\tau$ is an $n$-simplex of the simplicial set $\mathscr {F}(C_0)$. For $0 \leq i \leq n$, let $\tau _{i}$ denote the composite map

$\Delta ^{i} \hookrightarrow \Delta ^{n} \xrightarrow { \tau } \mathscr {F}(C_0) \rightarrow \mathscr {F}(C_ i).$

We then have a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \Delta ^{0} \ar [d]_{\tau _0} \ar@ {^{(}->}[r] & \Delta ^{1} \ar [d]^-{\tau _1} \ar@ {^{(}->}[r] & \Delta ^{2} \ar@ {^{(}->}[r] \ar [d]_{\tau _2} & \cdots \ar [d] \ar@ {^{(}->}[r] & \Delta ^{n} \ar [d]_{\tau _ n} \\ \mathscr {F}(C_0) \ar [r] & \mathscr {F}(C_1) \ar [r] & \mathscr {F}(C_2) \ar [r] & \cdots \ar [r] & \mathscr {F}(C_ n). }$

Consequently, we can view the pair $( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ as an $n$-simplex of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. The construction $(\sigma , \tau ) \mapsto ( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ determines a morphism of simplicial sets $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. In the special case where $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is a diagram of $\infty$-categories, we will refer to $\lambda _{t}$ as the taut scaffold of the cocartesian fibration $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Remark 5.3.4.12. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Then the diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \ar [rr]^{\lambda _{t}} \ar [dr] & & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & }$

commutes, where $\lambda _{t}$ is the morphism of Construction 5.3.4.11 and the vertical morphism are the projection maps of Construction 5.3.2.1 and Definition 5.3.3.1.

Example 5.3.4.13. Let $X$ be a simplicial set, which we identify with a diagram $\mathscr {F}: [0] \rightarrow \operatorname{Set_{\Delta }}$. Then the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ and the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ can both be identified with $X$ (see Examples 5.3.2.2 and 5.3.3.2). Under these identifications, the taut scaffold $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ of Construction 5.3.4.11 corresponds to the identity map $\operatorname{id}_{X}$.

Remark 5.3.4.14 (Functoriality). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the morphism of Construction 5.3.4.11. If $T: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is any functor between categories, then $\lambda$ induces a morphism

$\lambda '_{t}: \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}).$

Setting $\mathscr {F}' = \mathscr {F} \circ T$, we can use Remarks 5.3.2.3 and 5.3.3.7 to identify $\lambda '_{t}$ with a morphism from the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' )$ to the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}')$. This morphism coincides with the map obtained by applying Construction 5.3.4.11 to the diagram $\mathscr {F}'$.

Example 5.3.4.15 (Comparison of Fibers). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the morphism of Construction 5.3.4.11. Combining Example 5.3.4.13 with Remark 5.3.4.14, we see that for every object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers

$\{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$

is an isomorphism of simplicial sets (under the identifications provided by Remark 5.3.2.3 and Example 5.3.3.8, it corresponds to the identity morphism $\operatorname{id}: \mathscr {F}(C) \rightarrow \mathscr {F}(C)$).

Example 5.3.4.16. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, which we identify with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{Set_{\Delta }}$. Then the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the mapping cylinder $(\Delta ^1 \times X) \coprod _{ (\{ 1\} \times X) } Y$ (Example 5.3.2.13), and the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}([1])$ can be identified with the relative join $X \star _{Y} Y$ (Example 5.3.3.13). Under these identifications, Construction 5.3.4.11 corresponds to a morphism of simplicial sets

$\lambda _{t}: (\Delta ^1 \times X) \coprod _{ (\{ 1\} \times X) } Y \rightarrow X \star _{Y} Y.$

Unwinding the definitions, we see that this map classifies the commutative diagram

5.27
$$\begin{gathered}\label{equation:comparison-map-over-edge} \xymatrix@R =50pt@C=50pt{ \emptyset \star _{X} X \ar [r] \ar [d] & \emptyset \star _{Y} Y \ar [d] \\ X \star _{X} X \ar [r] & X \star _{Y} Y. } \end{gathered}$$

In particular, the morphism $\lambda _{t}$ is an isomorphism if and only if (5.27) is a pushout square of simplicial sets.

Proposition 5.3.4.17. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty$-categories indexed by a category $\operatorname{\mathcal{C}}$. Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Construction 5.3.4.11 is a scaffold of the cocartesian fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.

Proof. Condition $(0)$ of Definition 5.3.4.2 follows from Remark 5.3.4.12, condition $(2)$ from Example 5.3.4.15, and condition $(1)$ from the characterization of $U$-cocartesian morphisms supplied by Proposition 5.3.3.15. $\square$

Corollary 5.3.4.18. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a cocartesian fibration of $\infty$-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$.

Proof. Using Proposition 4.1.3.2, we can choose a diagram of $\infty$-categories $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \mathscr {F}'$. We can then take $\lambda$ to be the composition $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow { \alpha } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \xrightarrow {\lambda _{t}} \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$, where $\lambda _{t}$ is the taut scaffold of Proposition 5.3.4.17. $\square$

Corollary 5.3.4.19. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty$-categories indexed by $\operatorname{\mathcal{C}}$, and let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Proposition 5.3.3.15. Then there exists a levelwise categorical equivalence from $\mathscr {F}$ to the strict transport representation $\operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$.

Proof. Combine Proposition 5.3.4.17 with Remark 5.3.4.10 (for a more precise statement, see Construction 7.5.3.3). $\square$

In certain cases, one can improve on Example 5.3.4.15.

Proposition 5.3.4.20. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Suppose that, for every morphism $u: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the image $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ is a left covering map (Definition 4.2.3.8). Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Construction 5.3.4.11 is an isomorphism.

Proof. Let $( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ be an $n$-simplex of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. We identify $\sigma$ with a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, and each $\tau _ i$ with an $i$-simplex of the simplicial set $\mathscr {F}(C_ i)$. We wish to show that there is a unique $n$-simplex $\tau$ of $\mathscr {F}(C_0)$ satisfying $\lambda _{t}(\sigma , \tau ) = ( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$. Note that, for this condition to be satisfied, the simplex $\tau$ must be a solution to the lifting problem

$\xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [d] \ar [r] & \mathscr {F}(C_0) \ar [d] \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r]^{ \tau _ n } & \mathscr {F}(C_ n). }$

Since the inclusion $\{ 0\} \hookrightarrow \Delta ^ n$ is left anodyne (Example 4.3.7.11), our assumption that the right vertical map is a left covering guarantees that this lifting problem has a unique solution $\tau : \Delta ^ n \rightarrow \mathscr {F}(C_0)$ (Corollary 4.2.4.6). This proves uniqueness. To prove existence, write $\lambda _{t}(\sigma , \tau ) = ( \sigma , \{ \tau '_{i} \} _{0 \leq i \leq n} )$. We wish to prove that $\tau _ i = \tau '_ i$ for $0 \leq i \leq n$. For this, we observe that both $\tau _ i$ and $\tau '_ i$ can be viewed as solutions to a common lifting problem

$\xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r]^{ \tau _ i(0) } \ar [d] & \mathscr {F}(C_ i) \ar [d] \\ \Delta ^{i} \ar@ {-->}[ur] \ar [r] & \mathscr {F}( C_ n ). }$

Since the inclusion $\{ 0\} \hookrightarrow \Delta ^ i$ is left anodyne (Example 4.3.7.11) and the right vertical map is a left covering, the solution to this lifting problem is uniquely determined (Corollary 4.2.4.6). $\square$

Example 5.3.4.21 (Set-Valued Functors). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram of sets, and let us abuse notation by identifying $\mathscr {F}$ with a diagram of discrete simplicial sets. Then the taut scaffold $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is an isomorphism. It follows that $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ can be identified with the nerve of the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Example 5.3.2.5).

Corollary 5.3.4.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor which carries each morphism of $\operatorname{\mathcal{C}}$ to an isomorphism of simplicial sets. Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Remark 5.3.4.12 is an isomorphism.

Corollary 5.3.4.23. Let $\operatorname{\mathcal{C}}$ be a groupoid and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a Kan complex.

Proof. Using Corollaries 5.3.4.22 and 5.3.3.16, we see that the map $U: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration. Since $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.2.4.2), it follows that $U$ is a Kan fibration (Corollary 4.4.3.8), so that $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is also a Kan complex (Remark 3.1.1.11). $\square$

Example 5.3.4.24 (Homotopy Quotients). Let $G$ be a group and let $BG$ denote the associated groupoid (consisting of a single object with automorphism group $G$). Let $X$ be a simplicial set equipped with an action of $G$, which we identify with a functor $\mathscr {F}: BG \rightarrow \operatorname{Set_{\Delta }}$. Applying Corollary 5.3.4.22, we obtain an isomorphism of simplicial sets $X_{\mathrm{h}G} \xrightarrow {\sim } \operatorname{N}_{\bullet }^{\mathscr {F}}(BG)$, where $X_{\mathrm{h}G}$ is the homotopy quotient of $X$ by the action of $G$ (Example 5.3.2.15). If $X$ is a Kan complex, then Corollary 5.3.4.23 guarantees that $X_{\mathrm{h}G}$ is also a Kan complex.