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I am going to make this question deliberately vague.

A category is concrete if its objects can be realized as sets with extra structure (in particular, it admits a faithful functor to Set).

The examples I know of non-concretizable categories (the homotopy category, Cat with naturally isomorphic functors identified) are obviously derived from forgetting structure of higher categories. Is there some sense in which, given any locally small category, the objects of that category can be realized themselves as (small) higher categories with extra structure, analogous to considering faithful functors to Set?

EDIT: I think my question is as Jeremy suggested below: Given a locally small category, is it always the homotopy category of a concrete $(\infty,1)$-category, for some reasonable definition of concrete?

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    $\begingroup$ If your categories aren't required to be locally small, any category which is not locally small fails to be concretizable for reasons that seem to me to have nothing to do with higher categories. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 18, 2012 at 22:24
  • $\begingroup$ Sorry I meant locally small. $\endgroup$
    – John Berman
    Commented Oct 19, 2012 at 1:25
  • $\begingroup$ You may be interested in the discussion here: golem.ph.utexas.edu/category/2011/02/concrete_categories.html Several definitions of concrete infinity categories are proposed. Perhaps you are asking which 1-categories occur as homotopy categories of concrete infinity categories? $\endgroup$
    – Jeremy Hahn
    Commented Oct 19, 2012 at 2:38
  • $\begingroup$ This seems very unlikely to me. It's kind of a coincidence that the standard example of non-concrete categories are homotopy categories--they just happen to be the most (only?) naturally occurring examples. Essentially, most categories you will ever encounter are either small or accessible; the only way "real life" large categories fail to be accessible is if they are secretly actually accessible higher categories. $\endgroup$
    – Eric Wofsey
    Commented Oct 19, 2012 at 16:38
  • $\begingroup$ Eric is there a very tight relation between accessibility and concreteness? I would be interested to know it. I think John's question is unsolvable without a better concept of concrete infinity category. The cafe discussion suggests it might be reasonable, for instance, to think of every infinity category as concrete. It seems that concreteness is one of the few classical notions that hasn't yet been suitably generalized to the infinity context. $\endgroup$
    – Jeremy Hahn
    Commented Oct 20, 2012 at 21:42

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If you require you category to be actually small (and not just locally small) then the answer is yes.

Suppose $C$ is a small category. We can then define the category $C'$ as follows:

  • Objects($C'$) = $\{ C/t: t \in$ Object(C)$\}$
  • Morphisms($C'$) are functors between the corresponding categories

There is then a functor $F:C \rightarrow C'$ where

  • $F(t) = C/t$ for any object $t$ of $C$

  • For any map $\alpha:s \rightarrow t$ in $C$, $F(\alpha):C/s \rightarrow C/t$ is the functor where:

    • When $\beta:p \rightarrow s$ is an object of $C/s$ then $F(\alpha)(\beta) = \alpha \circ \beta:p \rightarrow t$ is the corresponding object of $C/t$.

    • Suppose $P:p\rightarrow s$ and $Q:q\rightarrow s$ are objects of $C/s$ and $g: P \rightarrow Q$ is a morphism in $C/s$ (i.e. a map $g:p \rightarrow q$ such that $q \circ g = p$).

      Then $F(\alpha)(\gamma) = g$ (as a map from $F(\alpha)(P) \rightarrow F(\alpha)(Q)$)

When dealing with locally small (but not necessarily small) categories however you have to be careful about set theoretic size issues.

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  • $\begingroup$ I believe that your functor is faithful making any small category concrete. $\endgroup$
    – Spice the Bird
    Commented Nov 2, 2012 at 16:59

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