Categories disguised as other structures

Question

It is common to hear that category theory unifies many apparently disparate areas of mathematics. One way it does so is by allowing us to take other mathematical structures and organize them into categories, then explore connections with other structures organized into categories and their opposites -- a famous example would be the equivalence between the category of Stone spaces and the opposite of the category of Boolean algebras.

A second way it allows us to unify other areas of mathematics is by permitting the definition of other mathematical objects inside a category with sufficient structure, like Frobenius algebras or groups inside a category, in a way that reproduces their standard set-theoretical definitions when realized in the category of sets but canonically adds structure to them when realized in other categories.

My question is not about either of the above methods of unification.

A third way in which categories unify many apparently disparate areas of mathematics is by allowing us to 'study them all at once' by realizing the objects of study in these fields as particular types of categories; a famous example here is that a group 'is' a category with one object, or that a preordered/partially ordered set 'is' a thin/skeletal thin category.

What are some other structures that can be realized as categories wearing a disguise?

To be a bit more specific, for another structure to 'be a category in disguise' I mean that we can define a correspondence between the (first or higher order) language of the other structure and the first order language of category theory such that the axioms of the other structure are immediately satisfied as a consequence of the axioms of a category, potentially with additional constraints like being thin or having products etc.

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As an aside, I don't mean to suggest that thinking about these structures as categories will allow a category theorist to instantly gain the insights of mathematicians in these other fields; those treasures are hard won and likely unavailable through other routes. What I hope is that some of the insight of categorical reasoning could be turned towards these structures, to complement the existing formidable mental architecture around them. Any contributions are appreciated.

Answer 1

Spaces (in the sense of homotopy theory). These have many Quillen equivalent models: the Serre–Quillen model structure on topological spaces, the Kan–Quillen model structure on simplicial sets, and the Thomason model structure on small categories.

It is the Thomason model structure that demonstrates that spaces in the sense of homotopy theory are (small) categories in disguise, up to a Thomason weak equivalence, i.e., a functor whose nerve is a weak homotopy equivalence.

This is not just a curiousity: the Thomason model structure is actively used in many fields that construct spaces out of categories, such as K-theory. Quillen's Theorems A and B and their generalizations are typical results formulated in this setting.

Answer 2

Graphs — in most of the various senses of that term.

Rather than your suggestion of “an interpretation of first-order theories”, I’ll focus primarily on “there’s a conservative, faithful (and perhaps full) functor from the (2-)category of widgets into $\mathrm{Cat}$”. Sometimes such a functor will be induced by an interpretation of first-order theories; but generally this notion allows for more general kinds of “disguise”.

Most generally, take “graphs” to mean the category-theorist’s sense, aka quivers, aka directed multigraphs with loops allowed. Taking the free category on such a graph gives a conservative and faithful functor from graphs to categories. Indeed, this is comonadic, and the coalgebra structure is unique when it exists; so graphs can be seen as categories with a certain property-like structure, and graph maps are functors that preserve this structure. This is discussed nicely here.

So graphs in this sense can certainly be seen as certain categories. But now most other standard notions of “graph” studied can be seen as certain instances of these ones:

• “no loops” is a property that can easily be imposed
• similarly, for (non-multi)-graphs, impose the property “parallel edges are equal”
• non-directed graphs can be viewed as directed graphs by replacing each non-directed edge $e\colon a \leftrightarrow b$ with a pair of directed edges $(e_1\colon a \to b;\; e_2 \colon a \leftarrow b)$. Under this, non-directed graphs can be seen as “graphs with a contravariant identity-on-objects involution, with no fixed points on arrows”. (“No fixed points on arrows” is needed for a correct treatment of loops.)

Answer 3

Algebraic theories can be identified with certain categories with finite products, a la Lawvere.

Answer 4

If you're willing to generalize from plain categories to enhanced types of categories like enriched/internal/indexed categories, then there are a lot more examples. Some famous ones are:

1. Sheaves on a site can be identified with certain categories enriched in an appropriate bicategory; see Walters, Sheaves on sites as Cauchy-complete categories.

2. Metric spaces can be identified with certain categories enriched over the poset $[0,\infty)$; see Lawvere, Metric spaces, generalized logic and closed categories.

3. Orbifolds can be identified with certain internal groupoids in smooth manifolds; see the references here.