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.

) 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 23:09in 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