I'm looking for, or hoping to inspire the creation of, a list of conventional names for categories that come up often.


For example, we have *the terminal category* $\fbox{$\bullet$}$, a nice name. I've heard this category $\fbox{$\bullet\to\bullet$}$ called *the free arrow category*. That's fine by me. What about the category with one object $x$ and an one arrow $p^n\colon x\to x$ for each natural number $n$ (and the obvious composition law)? It's hard to draw it without a package; here's my best attempt: $\fbox{$\bullet\circlearrowleft$}$. I'd like to call this *the free loop category*. But is that standard? 


The categories $[n]$ for $n\in\mathbb{N}$ might be drawn $\fbox{$\bullet^0\to\bullet^1\to\cdots\bullet^n$}$. I might call this *the length-$n$ chain category.* What about $\fbox{$\bullet\rightrightarrows\bullet$}$? I might call this *the parallel arrows category*. Would one know what I meant by *the two equalized arrows category* or *the two coequalized arrows category*? Hint: they each have three objects and four non-identity morphisms. But what if I didn't want a certain commutative diagram to hold there, i.e. I wanted to name the related five-morphism categories?


The point I hope is clear. Does anyone know of a definitive list of names for important diagram categories? 


Thanks!



Conventional names for finite categories
I notice that the categories considered for naming here are all the domains, or shapes, of basic *diagrams*; an object, an arrow, an endomorphism (n.b., my instinct was just to call that $\mathbb{N}$), a composable sequence, parallel arrows, equalized arrows... Not that diagrams *in* these categories aren't also interesting (a composable sequence in a composable sequence category, e.g., is well-worth half-an-hour's thought), but as diagram domains is where they all come up first for most of us; so why not call them what they are?


* $\fbox{$\phantom{X}$}$, the trivial diagram domain/the shape of the empty diagram
* $\fbox{$\bullet$}$, the *object* diagram domain/the shape of an object
* $\fbox{$\overset\bullet\circlearrowleft$}$, the endomorphism diagram domain/the shape of an endomorphism...


Of course, to establish a convention, one must write a famous textbook. Good luck with that!


