to the category Set. We say a category is concretizable if and only if it admits a faithful functor .
Very often it is useful to consider the case where is representable by some object , in that . For example, this is important for the statement of various concrete dualities induced by dual adjunctions. We say in this case that is representably concrete. By definition, the object is then a generator of the category.
One can also consider concrete categories over any base category instead of necessarily over . This is the approach taken in The Joy of Cats. Then the (small) categories concrete over form a 2-category .
The following furnish examples of concrete categories, with the first three representably concrete:
itself with generator the singleton set.
with the generator taken to be the one-point space.
Any monadic functor is faithful (because it preserves equalizers and reflects isomorphisms) and has a left adjoint. As special cases, we have the usual collection of examples of concrete categories: monoids, groups, rings, algebras, etc.
A category may be concretizable in more than one way:
Take to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by (so for all in the source). Then there are two versions of that one may use: one where (for a Banach space) consists of every vector in , and one where consists of those vectors bounded by (so the closed unit ball in ). The first may seem more obvious at first, but only the second is representable (by a -dimensional Banach space).
Insofar as categories such as , , , etc. admit many generators, these categories may be rendered representably concrete in a variety of ways. Indeed, the category may be monadic over in many different ways. For example, if is -dimensional, the functor is monadic and realizes as equivalent to the category of modules over the matrix algebra .
Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If is the category of sheaves on a small site , we have a familiar string of faithful functors
But if for example is the category of sheaves over , then no object can serve as a single generator of , since it cannot detect differences between arrows whenever the support of is strictly contained in the support of .
Every small category is concretizable (since it fully and faithfully embeds in the concrete category ).
If is concretizable, so is .
By assumption, there is a faithful functor , and is monadic.
Of course, since a category may possess a generator but no cogenerator, it does not follow that is representably concrete if is.
“Only if” was proven in (Isbell). To prove it, note that if is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that is the equalizer of , and is the equalizer of . If as subobjects of , then since and so , we must also have ; hence (since is faithful) , so that factors through . Conversely, factors through , so we have as subobjects of . Since is regularly well-powered, it follows that any category admitting a faithful functor to must also be so.
(Actually, Isbell proved a more general condition which applies to categories that may lack finite limits.)
A relatively deep application of Isbell’s result is that the classical homotopy category Ho(Top) of topological spaces is not concretizable, even though it is a quotient of which is concretizable. (Freyd 70)
Peter Freyd, Concreteness, JPAA 3 (1973)
Peter Freyd, Homotopy is not concrete, in The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168, Springer-Verlag, 1970, Republished in: Reprints in Theory and Applications of Categories, No. 6 (2004) pp 1-10 (web)