I posted this on [math.stack.exchange](https://math.stackexchange.com/) but didn't get a helpful response, so please let me try it here.


Let $D$ be a small category and $F:D\to sSets$ a functor.


There is a bisimplicial set indicated by 
$$
...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod\_{d\_1\to d\_0}F(d\_1)\begin{array}{c}\to\\\to\end{array}\coprod\_{d\_0}F(d\_0)
$$
which i like to call $sres(F)\_{\bullet\bullet}$. By considering either the horizontal or the vertical index first, I get two functors $H:\Delta^{op}\to sSets$ and $V:\Delta^{op}\to sSets$ (the ''vertical'' direction is the one of the simplicial sets $F(d)$, i.e. each object in the diagram above should be imagined as a vertical column).


The [homotopy colimit](http://ncatlab.org/nlab/show/homotopy+limit#homotopy_colimits_over_diagrams_of_spaces_91) $\operatorname{hocolim}F$ of $F$ can be defined in different ways (up to weak equivalence) and one way is $(\operatorname{hocolim}F)\_n=sres(F)\_{n n}$, the diagonal of $sres(F)\_{\bullet\bullet}$.



> 
> Is there a weak equivalence between between $\operatorname{hocolim}F$ and $\operatorname{hocolim}H$? (I don't think there is a weak equivalence between $\operatorname{hocolim}F$ and $\operatorname{hocolim}V$, or is it?)
> 
> 
> 


Let me call a simplicial set $A$ *$k$-dimensional*, if it is equal to its $k$-[skeleton](http://ncatlab.org/nlab/show/simplicial+skeleton) $\mathbf{sk}\_k$ (If I am not mistaken, $k$ should be the smallest integer such that all elements of $A\_{k+1}$ are degenerated simplices).



> 
> Suppose that the [nerve](http://ncatlab.org/nlab/show/nerve) $N(D)$ of $D$ is weakly equivalent to a simplicial set $A$ of dimension $k$. Is it true that $\operatorname{hocolim}F$ is weak equivalent to the homotopy colimit of the diagram
>  $$
> \coprod\_{d\_{k+1}\to ...\to d\_0}F(d\_{k+1})\begin{array}{c}\to \\ \vdots\\\to\end{array}...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod\_{d\_1\to d\_0}F(d\_1)\begin{array}{c}\to\\\to\end{array}\coprod\_{d\_0}F(d\_0),
> $$
>  (the truncated $H$) or in other words, can I ''stop'' at the $k+1$th stage of the diagram in the horizontal direction to calculate the homotopy colimit? If yes, why?
> 
> 
> 


In particular, if $D$ is linear (for example $\mathbb{N}$ or $\cdot\to\cdot\to\cdot$), this would mean, that $\operatorname{hocolim}F$ is the homotopy colimit of the simple diagram
$$
\coprod\_{d\_1\to d\_0}F(d\_1)\begin{array}{c}\to\\\to\end{array}\coprod\_{d\_0}F(d\_0).
$$




---


My third question is a little vague and like the second one but not ''for the source'' of $F$ but ''for the target''. Please don't hesitate to post an answer only to the first two questions, if this third one is not well formulated. I wondered, why one takes only two-limits of stacks and not $k$-limits. The nerve of a category is a [2-coskeletal](http://ncatlab.org/nlab/show/simplicial+skeleton#examples_20) simplicial set and this is where the reason comes from, I guess.



> 
> Suppose the functor $F$ factorize through the nerve $F:D\to CAT\xrightarrow{N} sSets$. Is it true that $\operatorname{hocolim}F$ is weakly equivalent to the homotopy colimit of the diagram
>  $$
> \coprod\_{d\_2\to d\_1\to d\_0}F(d\_{2})\begin{array}{c}\to \\ \to\\\to\end{array}\coprod\_{d\_1\to d\_0}F(d\_1)\begin{array}{c}\to\\\to\end{array}\coprod\_{d\_0}F(d\_0)
> $$
>  and what is the reason? If not, what did I misunderstand here?
> 
> 
> 



Three questions on $\operatorname{hocolim}$
**First question**$\newcommand{\op}[1]{{#1}^{\mathrm{op}}}$$\newcommand{\sSet}{\mathrm{sSet}}$$\newcommand{\Grpd}{\mathrm{Grpd}}$$\newcommand{\Cat}{\mathrm{Cat}}$$\newcommand{\NN}{\mathbb{N}}$$\newcommand{\sres}{\mathrm{sres}}$$\newcommand{\hocolim}{\operatorname{hocolim}}$$\newcommand{\diag}{\operatorname{diag}}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\real}[1]{\lvert #1 \rvert}$


The homotopy colimit of a functor $X:\op{\Delta}\to\sSet$ is equivalent to its geometric realization $\real{X}$. This holds because every functor $\op{\Delta}\to\sSet$ is cofibrant in the Reedy model structure. In particular, geometric realization of simplicial simplicial sets preserves objectwise weak equivalences. You may consult corollary 15.8.8 and theorem 18.7.4 in Hirschhorn's book "*Model categories and their localizations*" for these results. The equivalence between the homotopy colimit and the geometric realization is given explicitly by the [Bousfield–Kan map](http://ncatlab.org/nlab/show/Bousfield-Kan+map) (in section 18.7 of Hirschhorn's book).


On the other hand, it is a very interesting well-known fact that the geometric realization of a simplicial simplicial set $X:\op{\Delta}\to\sSet$ is canonically isomorphic to the diagonal $\diag X$ of the associated bisimplicial set. You will find this as theorem 15.11.6 of Hirschhorn's book, or as exercise IV.1.4 in the book "Simplicial homotopy theory\*" by Goerss and Jardine. [Further, if you are very categorically inclined, you may see this quite formally as a consequence of geometric realization being a coend whose weight is the Yoneda embedding.]


From the preceding paragraphs, we conclude that there is a natural weak equivalence $\hocolim X\overset{\sim}{\To}\diag X$ for every simplicial simplicial set $X$. This is stated explicitly as corollary 18.7.7 of Hirschhorn's book.


What I said so far is in part briefly described in section XII.3.4 of the book "Homotopy limits, completions and localizations" by Bousfield and Kan.



> 
> **Answer to first question:** We conclude from the previous discussion that the homotopy colimits of both $H$ and $V$ are naturally equivalent to their diagonals. Each of these diagonals is by definition also the diagonal of $\sres(F)$, i.e. your proposed definition for the homotopy colimit of $F$.
> 
> 
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[As a consistency check, the definition of homotopy colimit of a diagram $F:D\to\sSet$ given in both Bousfield–Kan and Hirschhorn is isomorphic to your definition as the diagonal of $\sres(F)$. You can see a brief explanation of this fact in section 4.13 of Dan Dugger's notes "[*A primer on homotopy colimits*](http://pages.uoregon.edu/ddugger/hocolim.pdf)". These are very readable notes, and I highly recommend them. You can also read about the Bousfield–Kan map in section 17.2 of those notes.]


**Second question**



> 
> The answer to your second question is *no*.
> 
> 
> 


Take the very simple category $D$ described schematically as $0\overset{a}{\to} 1\overset{b}{\to} 2$. Then the category $D$ has a terminal object so its nerve is contractible, and we may take $k=0$ in your second question. Consider the terminal functor $F=\ast : D\to \sSet$ whose value is always the terminal simplicial set with a single vertex, and no non-degenerate higher dimensional simplices. The homotopy colimit of this diagram is equivalent to the nerve of $D$, and thus contractible. On the other hand, the $1$-truncated simplicial object you write in your question is levelwise discrete, so we can think of it as a simplicial set with:


* three vertices $0$, $1$, and $2$
* three non-degenerate edges: $a$ from $0$ to $1$, $b$ from $1$ to $2$, and $ab$ from $0$ to $2$.


Non-coincidentally, this is just the $1$-skeleton of the nerve of $D$. It is quite easy to see that its realization is a circle. Thus the homotopy colimit of your $1$-truncated simplicial object is not contractible. The problem is of course that we "erased" the non-degenerate $2$-simplex from the nerve of $D$, thus changing its homotopy type.



> 
> On a positive note, I believe a formula *like* the one you state does hold. Under the stronger assumption that the nerve of $D$ is $k$-skeletal, then the homotopy colimit of $F:D\to\sSet$ is equivalent to the homotopy colimit of a $k$-truncated simplicial object like the one you write (except you write a $(k+1)$-truncated simplicial object instead). [By a $k$-truncated simplicial object, I mean a functor indexed by the full subcategory of $\op{\Delta}$ spanned by the $n$-simplices with $n\leq k$.]
> 
> 
> 


*Sketch of proof*: If the nerve of $D$ is $k$-skeletal, $\sres(F)$ is also (horizontally) $k$-skeletal. Consequently, the diagonal of $\sres(F)$ coincides with the geometric realization of the horizontal $k$-truncation of $\sres(F)$ (in your question you actually write down the $(k+1)$-truncation). This is equivalent to the homotopy colimit of the $k$-truncation for the same reasons as before: every $k$-truncated simplicial object in simplicial sets is Reedy cofibrant (which actually follows from the analogous result for simplicial simplicial sets).


**Intermezzo**


I will make a couple of simple comments regarding your examples in between the second and third questions.


The homotopy colimit of any functor $F$ indexed by a category $D$ with a terminal object $d$ is weakly equivalent to $F(d)$. So the homotopy colimit of any functor $F$ indexed by $0\to 1\to 2$ is weakly equivalent to $F(2)$.


Moreover, any filtered colimit in simplicial sets is actually a homotopy colimit, essentially because filtered colimits commute with homotopy groups in $\sSet$. In other words, homotopy colimits in $\sSet$ along filtered categories (such as $\NN$) are no harder than ordinary colimits.


**Third question**


For this question, the relevant part about stacks is not that they have values in categories. It is that they have values in *groupoids*. Moreover, one needs to take the homotopy colimit *in groupoids* and not in simplicial sets. I will elaborate below.


First, here is a counter-example to your third question as it stands. Consider any category $C$ whose nerve is equivalent to $S^3$, for example. [Any simplicial set is weakly equivalent to the nerve of some category, for example its category of simplices. Here, however, is an explicit example for $S^3$: simply take the [join of categories](http://ncatlab.org/nlab/show/join+of+categories) $C = 2\ast 2 \ast 2 \ast 2$, where $2$ denotes a discrete category with two objects.] Then the homotopy colimit along $C$ of the nerve of the constant terminal functor $\ast:C\to\Cat$ is simply
$$ \hocolim N(\ast) = \hocolim \ast \simeq BC \simeq S^3 $$
However, the homotopy colimit of the corresponding $2$-truncated simplicial set you write will be equivalent to its geometric realization, which is $2$-dimensional. In particular, it cannot have non-trivial homology in dimension $3$, and cannot be equivalent to $S^3$.


What is true is that the homotopy colimit *in groupoids* (with respect to the [natural model structure](http://ncatlab.org/nlab/show/canonical+model+structure+on+groupoids)) of a functor $F:D\to\Grpd$ (from $D$ to the category of groupoids) is equivalent to the homotopy colimit *in groupoids* of the $2$-truncated simplicial object you write down.


If you want to further compose $F:D\to\Grpd$ with the nerve functor, then you will have to apply the fundamental groupoid functor to the homotopy colimit of $N\circ F$ in simplicial sets to recover the homotopy colimit of $F$ in groupoids. This holds because the fundamental groupoid functor is part of a Quillen adjunction between simplicial sets and groupoids. The same holds for the homotopy colimit of the $2$-truncated simplicial object you write down.


