Set theory has a very convenient and well established curly brace notation to specify a set by its elements: $\{2,3,4,6\}$ or $\{\text{finite subgroups of }SU(2)\}$ are simple examples.

There should be a similar convenient notation for specifying a category by its objects and morphisms. Such a notation should easily accommodate categorical constructions such as slice categories. For example a double slash notation to separate objects and morphisms would define a slice category by something like (I am making this up!) $$ \mathcal{C}\downarrow X= [ Y\to^f X : [Y//f]\in \mathcal C\quad // \quad (Y\to^f X)\to^h (Z\to^g X): [Y,Z//h] \in \mathcal C, g\circ h=f] $$ (a commutative diagram in the second part of the specification would be more convenient here, but it should also be possible to typeset the notation inline).

Do such notations already exist? Whether they do or not, what notations would contributors recommend or suggest?

Update. Many thanks for comments made here. So far I most like the observation that clearly describing the morphisms makes the objects implicit. Still, I think beginners need the objects too, and have been experimenting with a notation like the one above, but using "staples" instead of square brackets, and introducing morphisms after objects by a vertical rectangular block (a bit like a closed staple). 2-morphisms could then be introduced in a similar way by a double block (a block with a vertical line through it). While the answers convince me that such notation is often unnecessary and maybe unhelpful sometimes, I'm not convinced such notation would be worthless.

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    $\begingroup$ I would advise against using double slash as you propose (especially $Y//f$) as this is a notation for a 'homotopy' or weak quotient in some settings. An example is $X//G$, with $X$ a $G$-set, $G$ a group, this is then the action groupoid associated to the action. $\endgroup$ – David Roberts Dec 12 '10 at 23:10
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    $\begingroup$ The Ehresmann school of category theory (which is very small, and I only know of a handful of living practitioners) named categories by their morphisms, because a lot of the time there was focus on categories with the same objects. By saying $[X/G](S)$ is $G$-equivariant maps between $G$-bundles over $S$ in $Sch/X$ (or similar, I can't recall the precise details of the stack $[X/G]$ at present) you are pretty much there. Personally I would work up to equivalence and say that this groupoid is (equivalent to) $Hom(S,X//G)$ in the 2-category of anafunctors internal to schemes, but that's just me $\endgroup$ – David Roberts Dec 13 '10 at 1:03
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    $\begingroup$ One more comment: describing a set by its elements is very uncategorial - as far as category theory is concerned a bare() set is fairly interchangable by one isomorphic to it, especially for finite sets. [()By a bare set I mean not one that is defined as a subset of some other given object, as in the subgroups example.] Do you care if you one-element set is {*} or {\empty} or {1}? Do you worry if this is accidentally contained in any other set you consider? $\endgroup$ – David Roberts Dec 13 '10 at 23:09
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    $\begingroup$ And, to add to Mike's suspiciousness... I do not know what changed since the 20th century that would justify the expectation that in the 21st century we ought to be able to describe simple examples of categories consicely. $\endgroup$ – Mariano Suárez-Álvarez Dec 13 '10 at 23:30
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    $\begingroup$ In 20-30 years time we might want to introduce categorical thinking more systematically at an undergraduate level (as set theory becomes an outdated foundation). For most mathematical purposes, I don't care what is the element of a 1-element set, nor do I care precisely what an ordered pair or a disjoint union is, as long as the universal properties are satisfied. I agree that describing sets by elements is uncategorical, but that applies to morphisms too, and we aren't going to introduce infinity categories in one breath are we? $\endgroup$ – David MJC Dec 14 '10 at 0:24

One thing I often do is work with set-builder notation $\{blah\in thing| conditions\}$ where either or both of the $blah$ and $conditions$ are allowed to be (collections of) (2-)commuting diagrams. I also work with objects and arrows separately. (Edit: by which I mean I write $Obj(C) := \ldots$ and $Mor(C) := \ldots$ or similar)

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    $\begingroup$ Can you give an example? Thanks $\endgroup$ – David MJC Dec 12 '10 at 23:20
  • $\begingroup$ How do you get the arrows in? $\endgroup$ – darij grinberg Dec 13 '10 at 13:01
  • $\begingroup$ Ah, well this is generally in my handwritten notes. In display environment you can have 'stretchy' braces and pop an \xymatrix or \array where needed. As far as fancy options for TeX-ing this notation, I recommend asking at tex.stackexchange.com $\endgroup$ – David Roberts Dec 13 '10 at 23:05

For finite categories it is customary to give a presentation in terms of generators and relations (equations), where the generators are presented as a directed graph whose vertices are the objects and the arrows are the generating arrows.

For many "schematic" categories, such as slice categories, it is customary to draw the relevant shapes of objects and morphisms. Extra conditions, if any, go in the text.

I do not quite understand why you think it should be possible to typset the convenient notation inline. Convenient notation for categories simply isn't one-dimensional.

  • $\begingroup$ Typesetting inline is not a requirement, but note that I could have typeset the commutative diagram inline if I had the energy! Two and three dimensional diagrams can often be represented concisely on paper, hence inside a "Category of..." notation. $\endgroup$ – David MJC Dec 12 '10 at 23:10

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