# Make category theory less abstract

It is known that every group is isomorphic to a permutation group. Is there similar result for categories? Or is it possible in principle? I.e. can we hope to make categories less abstract?

• en.wikipedia.org/wiki/Concrete_category .... See in particular further examples no. 6, which uses the Yoneda embedding to find a faithful functor from a small category C to SET. Also see the interesting counter example on wikipedia, which is hTop. Jul 4, 2017 at 19:02
• Why the votes to close? I think this is a fine question. Jul 4, 2017 at 20:21
• Then we'd have to call it concrete nonsense :-) Jul 4, 2017 at 21:33

There is a very direct analog: you can define a notion of a "category action" on a set, and every small category has a Cayley representation, acting on its set of arrows by composition.

It turns out, however, that the category of left $\mathcal{C}$-sets is equivalent to the category of functors $\mathcal{C} \to \mathbf{Set}$, and the category of right $\mathcal{C}$-sets is equivalent to the category of functors $\mathcal{C}^\text{op} \to \mathbf{Set}$; the usual formulation of category theory already emphasizes functor categories, especially $\mathbf{Set}$-valued functor categories, so there is not much benefit to introducing the notion of a category action as a separate notion.

Under this equivalence, the Cayley representation of a category corresponds to the yoneda embedding.

(that said, category actions do become very important in internal category theory, as a substitute for the harder-to-define notion of a functor from an internal category to the ambient category)

In any case, all of this leads to a canonical way to make any small category into a concrete category. Its "underlying set" functor $U$ is given by:

• To an object $X$, $U(X)$ is the set of all arrows with codomain $X$
• To a morphism $f:X \to Y$, $U(f)$ is the function $U(X) \to U(Y)$ given by composition with $f$. (i.e. it sends $x : U \to X$ to $f \circ x : U \to Y$)

If you choose a family of generators, an alternative is to restrict $U(X)$ to just those arrows whose domain is in the generating set. In this way you can adapt this construction to any locally small category with a small family of generators.

However, rather than going through this device to make a category into a concrete category, it is more common to instead either:

• Work with the yoneda embedding
• Introduce the notion of generalized elements, and learn how to adapt element-like reasoning to work with generalized elements
• Vaughan Pratt in The Yoneda Lemma without Category Theory does take an perspective that emphasizes "category actions" which he calls "(heterogeneous) modules". The overall approach may well be in the vein the OP is thinking. Jul 4, 2017 at 23:06