David Madore asked about non-concreteness > > So do you mean it is just a question of cardinality? Does the > homotopy category become concrete if we limit it to topological spaces > (and maps and homotopies) belonging to some sufficiently closed > universe of set theory (like a V_alpha where alpha is an ordinal of > sufficiently large cofinality)? > > Or, to ask a simpler question to start with: is every finite category > concret(izabl)e?
The cardinality argument I gave earlier is just a motivation, not a correct proof, as I said.
But yes, every category defined in some universe of sets does become concrete in some larger universe. The construction is like the one for finite catgories that Tobias Fritz gave. It is the Yoneda embedding. Very roughly you replace each object A of the category by the "set" of all arrows to A. Then an arrow f:A-->B acts as a function taking all the arrows g:C-->A (where C may be any onject of the category) to the composites fg:C-->B. More precisely you replace A by the set-valued functor h_A represented by A, and each arrow f:A-->B by a natural transformation h_f between the functors h_A-->h_B. This is the Yoneda functor or Yoneda embedding and correct definitions of it are in many books and probably many web sites.
But of course when you take care with foundations there is, for example, no set of all group homomorphisms from arbitrary groups to a given group G. There is a proper class of them, just because there is a proper class of groups (even of non-isomorphic groups). If the category you started with was finite this would be no issue. And even when it is much larger there are often technical tricks to get around it. But for the general case you have to start with a category in some Grothendieck universe U and be willing to ascend to a larger universe U' with U as a member.