Conventional names for finite categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:53:50Z http://mathoverflow.net/feeds/question/104206 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104206/conventional-names-for-finite-categories Conventional names for finite categories David Spivak 2012-08-07T15:15:34Z 2012-08-07T18:44:54Z <p>I'm looking for, or hoping to inspire the creation of, a list of conventional names for categories that come up often.</p> <p>For example, we have <em>the terminal category</em> $\fbox{$\bullet$}$, a nice name. I've heard this category $\fbox{$\bullet\to\bullet$}$ called <em>the free arrow category</em>. 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 <em>the free loop category</em>. But is that standard? </p> <p>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 <em>the length-$n$ chain category.</em> What about $\fbox{$\bullet\rightrightarrows\bullet$}$? I might call this <em>the parallel arrows category</em>. Would one know what I meant by <em>the two equalized arrows category</em> or <em>the two coequalized arrows category</em>? 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?</p> <p>The point I hope is clear. Does anyone know of a definitive list of names for important diagram categories? </p> <p>Thanks!</p> http://mathoverflow.net/questions/104206/conventional-names-for-finite-categories/104209#104209 Answer by Mike Shulman for Conventional names for finite categories Mike Shulman 2012-08-07T16:14:00Z 2012-08-07T16:14:00Z <p>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.</p> http://mathoverflow.net/questions/104206/conventional-names-for-finite-categories/104211#104211 Answer by Mariano Suárez-Alvarez for Conventional names for finite categories Mariano Suárez-Alvarez 2012-08-07T16:47:02Z 2012-08-07T16:52:10Z <p>The two-parallel-arrow category should be called the Kronecker category (it is called the Kronecker quiver by representation theorists)</p> <p>I'd like the loop to be called <em>Jordan category</em>, for its representation theory is that of Jordan canonical forms —but plain ol' "the loop" is good enough :-)</p> <p>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. &amp;c.</p> http://mathoverflow.net/questions/104206/conventional-names-for-finite-categories/104220#104220 Answer by some guy on the street for Conventional names for finite categories some guy on the street 2012-08-07T18:44:54Z 2012-08-07T18:44:54Z <p>I notice that the categories considered for naming here are all the domains, or shapes, of basic <em>diagrams</em>; 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 <em>in</em> 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?</p> <ul> <li>$\fbox{$\phantom{X}$}$, the trivial diagram domain/the shape of the empty diagram</li> <li>$\fbox{$\bullet$}$, the <em>object</em> diagram domain/the shape of an object</li> <li>$\fbox{$\overset\bullet\circlearrowleft$}$, the endomorphism diagram domain/the shape of an endomorphism...</li> </ul> <p>Of course, to establish a convention, one must write a famous textbook. Good luck with that!</p>