# Conventional names for finite categories

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!

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My collaborators Tom Church, Benson Farb and I recently wrote a paper about the category of finite sets with inclusions. We couldn't find a standard name for it so we called it "FI" in our paper, but it turns out that people talk about it under an endless variety of names. The area is not Linneaeized, to say the least. – JSE Aug 7 '12 at 15:25
The free loop category could also be called the free dynamical system. – Qiaochu Yuan Aug 7 '12 at 16:46
I would call the loop category $B \mathbb{N}$ by analogy with $BG$. – Eugene Lerman Aug 7 '12 at 17:46
I second Eugene's comment (possibly calling it $\mathbf{B}\mathbb{N}$, though), and add that analogous names exist for other monoids, such as $\mathbb{Z}/n$. – David Roberts Aug 7 '12 at 23:24

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!

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This is basically the same philosophy behind names of the form "the free X category". – Mike Shulman Aug 12 '12 at 20:31
True. Choosing one convention or the other is a matter of poetics; the question struck me as being one of poetics at the beginning, anyway. However: it seems that one shouldn't call any category a "free coequalizer", because the thing that comes to mind admits functors that aren't coequalizer of anything, but it nearly is "the shape of a coequalizer" in a sensible way. This distinction may be artificial, however, since being a coequalizer is a property, not just stuff. – some guy on the street Aug 13 '12 at 15:54

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.

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So is the loop the Jordan category or the natural numbers. – Benjamin Steinberg Aug 7 '12 at 18:28
I'm leery of naming small things after people, even though I do still remember what the Klein V(ier) group and the Sierpinski Space are (the one that isn't a gasket or a carpet --- why "gasket", though?); for some reason I don't mind putting people's names on big things, like "Hilbert" for the category of Hilbert spaces, and on theorems, too. But what on paper is "Kronecker" about this Kronecker quiver? Or, why is "Jordan canonical forms" the correct filter to view endomorphisms through? – some guy on the street Aug 7 '12 at 18:54
Kronecker classified all functors from this category to fd vector spaces. Since these are the only functors rep theorists care about they use the Kronecker name. Ditto for the loop. – Benjamin Steinberg Aug 7 '12 at 19:37
@Benjamin, the point I was trying to make somewhat elliptically, really, that there is something like the particle-wave duality that applies to these categories: what you call them depends on what you are doing :-) – Mariano Suárez-Alvarez Aug 7 '12 at 19:48
(Rep theorists have considered infinite dimensional reps of the Kronecker quiver, by the way!) – Mariano Suárez-Alvarez Aug 7 '12 at 19:58

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.

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I guess what would be nice is a database of names. In other words, a function from a set of names of categories to a set of categories. In fact, it would be nice to extend this database to include names of functors as well. Hell, why not go all the way: Categories and functors could be classified into types (finite categories, op-fibrations, etc.) and well-known properties could be recorded there. I think this would be useful. – David Spivak Aug 7 '12 at 22:07
@David - as usual this was already thought of, and discussed at the nForum. But with your categorical database-fu you may like to help (and bask in the irony). – David Roberts Aug 7 '12 at 23:25