Conventional names for finite categories - MathOverflow

Conventional names for finite categories

2012-08-07T15:15:34Z

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!

Answer by Mike Shulman for Conventional names for finite categories

2012-08-07T16:14:00Z

You're definitely not going to get consensus. I've heard your "free arrow category" also called "the walking arrow" and "the arrow category" and "the interval category" and "the directed interval category" and "the ordinal 2", just off the top of my head. The length-n chain category is, I think, a bit more commonly called "the ordinal $n+1$". Your other names seem reasonable, but I've never heard of any standard list, so I would just define each name at the point of first use.

Answer by Mariano Suárez-Alvarez for Conventional names for finite categories

2012-08-07T16:47:02Z

The two-parallel-arrow category should be called the Kronecker category (it is called the Kronecker quiver by representation theorists)

I'd like the loop to be called Jordan category, for its representation theory is that of Jordan canonical forms —but plain ol' "the loop" is good enough :-)

For categories which are posets, the "correct" name is that of the poset, I think; this covers the paths, for example (but it sort of sucks that the ordinal $n$ is what we usually want to index $n-1$...). For categories which are actually monoids (like your example with one object and $\mathbb N_0$ as arrows) should be called by the name of the monoid. &c.

Answer by some guy on the street for Conventional names for finite categories

2012-08-07T18:44:54Z

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!