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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?

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    $\begingroup$ 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. $\endgroup$
    – Elle Najt
    Jul 4, 2017 at 19:02
  • $\begingroup$ Why the votes to close? I think this is a fine question. $\endgroup$ Jul 4, 2017 at 20:21
  • $\begingroup$ Then we'd have to call it concrete nonsense :-) $\endgroup$
    – user4894
    Jul 4, 2017 at 21:33

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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
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