In what follows, all categories are assumed to be small (classes of objects and morphisms are sets).

Which mathematical structures $X$ can be seen as particular cases of small categories $\underline{X}$, so that the morphisms $X\to Y$ of those structures coincide with functors $\underline{X}\to \underline{Y}$?

So far, I have:

  • set $S$ (objects are elements of $S$, there are no morphisms except identities); a set is precisely a category without non-identity morphisms;
  • semigroup $S$ (one object, morphisms are elements of $S$, composition is multiplication in $S$); a unital semigroup is precisely a category with just one object; a group is precisely a category with one object and all morphisms invertible;
  • multidigraph $\Gamma$ (objects are vertices of $\Gamma$, morphisms are directed paths, composition is concatenation); a multidigraph is precisely a free category;
  • posets $P$ (objects are elements of $P$, morphisms are elements of $\leq$); posets are precisely categories with at most one arrow between any two objects).

Basically I'm asking is

What known (popular) categories are full subcategories of $\textsf{Cat}$, the category of all small categories and functors? How about topological spaces? Rings? Modules? Lattices? Simplicial complexes?

Is the category algebra studied for such cases?

I'm beginning to realize that associative algebras seem to be the central concept through which all others are studied, such as groups (group algebras), rings ($\mathbb{Z}$-algebras), ideals (associated graded algebras), modules (tensor, symmetric, exterior algebras), Lie algebras (universal enveloping algebras), simplicial complexes (Stanley-Reisner algebras), small categories (categorical algebras such as semigroup, incidence, quiver algebras), topological spaces (algebra of continuous functions to $\mathbb{R}$), affine and projective algebraic sets (coordinate algebras). Thus I'm wondering how much the category algebras really cover.

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    I have to nit-pick on the issue of directed multigraphs. Concatenation of paths is something external to the graph as such, which are just a pair of sets and a pair of unary functions. – Malice Vidrine May 28 '14 at 18:58
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    Some people study simplicial complexes without studying Stanley-Reisner algebras! – Qiaochu Yuan May 28 '14 at 19:07
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    Also, a graph isn't a free category in the sense that a functor between free categories is more general than a morphism between graphs. So you don't get a full subcategory this way. – Qiaochu Yuan May 28 '14 at 19:08
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    Rings are $\text{Ab}$-enriched categories with one object. Lattices are posets with an extra property (not an extra structure). Simplicial complexes are essentially their face posets. – Qiaochu Yuan May 28 '14 at 19:10
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    I also think you're overgeneralizing. Associative algebras are great but they aren't that great. It's convenient to attach algebras to things but there are other things you can do. – Qiaochu Yuan May 28 '14 at 19:15

The category of groupoids is (by definition) a full subcategory of $\mathsf{Cat}$. There are many natural examples of groupoids. Besides, groupoids really lie on the intersection between algebra and topology. For example, $\infty$-groupoids should correspond to the homotopy types of spaces.

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