nLab
concrete category

Contents

Idea

A concrete category is a category that looks like a category of “sets with extra structure”, that is a category of structured sets.

Definition

Definition

A concrete category is a category CC equipped with a faithful functor

U:CSet U : C \to Set

to the category Set. We say a category CC is concretizable if and only if it admits a faithful functor U:CSetU: C \to Set.

Remark

Very often it is useful to consider the case where UU is representable by some object c 0Cc_0 \in C, in that UC(c 0,)U \simeq C(c_0,-). For example, this is important for the statement of various concrete dualities induced by dual adjunctions. We say in this case that (C,U:CSet)(C, U: C \to Set) is representably concrete. By definition, the object c 0c_0 is then a generator of the category.

We remark that the existence of a left adjoint FF to U:CSetU: C \to Set implies that UU is representable by F(1)F(1). Conversely, if CC has coproducts or even just copowers, then representability of UU implies that UU has a left adjoint.

Remark

One can also consider concrete categories over any base category XX instead of necessarily over SetSet. This is the approach taken in The Joy of Cats. Then the (small) categories concrete over XX form a 2-category Cat(X)Cat(X).

Examples

The following furnish examples of concrete categories, with the first three representably concrete:

  • C=SetC = Set itself with generator c 0={}c_0 = \{\bullet\} the singleton set.

  • C=TopC = Top with the generator c 0c_0 taken to be the one-point space.

  • Any monadic functor U:CSetU: C \to Set 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 C C to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by 1 1 (so Tvv \| T v \| \leq \| v \| for all v v in the source). Then there are two versions of U U that one may use: one where U(V) U ( V ) (for V V a Banach space) consists of every vector in V V , and one where U(V) U ( V ) consists of those vectors bounded by 1 1 (so the closed unit ball in V V ). The first may seem more obvious at first, but only the second is representable (by a 1 1 -dimensional Banach space).

  • Insofar as categories such as SetSet, TopTop, Vect kVect_k, etc. admit many generators, these categories may be rendered representably concrete in a variety of ways. Indeed, the category Vect kVect_k may be monadic over SetSet in many different ways. For example, if VV is nn-dimensional, the functor hom(V,):Vect kSet\hom(V, -): Vect_k \to Set is monadic and realizes Vect kVect_k as equivalent to the category of modules over the matrix algebra hom(V,V)\hom(V, V).

  • Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If E=Sh(C,J)E = Sh(C, J) is the category of sheaves on a small site (C,J)(C, J), we have a familiar string of faithful functors

    Sh(C,J)Set C opmonadicSet/C 0Σ C 0Set.Sh(C, J) \hookrightarrow Set^{C^{op}} \stackrel{monadic}{\to} Set/C_0 \stackrel{\Sigma_{C_0}}{\to} Set.

    But if for example EE is the category of sheaves over \mathbb{R}, then no object XX can serve as a single generator of EE, since it cannot detect differences between arrows YZY \stackrel{\to}{\to} Z whenever the support of YY is strictly contained in the support of XX.

  • A concrete category that is equipped with the structure of a site in a compatible way is a concrete site. The category of concrete sheaves on a concrete site is concrete.

Properties

Proposition

Every small category CC is concretizable (since it fully and faithfully embeds in the concrete category Set C opSet^{C^{op}}).

Proposition

If CC is concretizable, so is C opC^{op}.

Proof

By assumption, there is a faithful functor U op:C opSet opU^{op}: C^{op} \to Set^{op}, and hom(,2):Set opSet\hom(-, \mathbf{2}): Set^{op} \to Set is monadic.

Remark

Of course, since a category CC may possess a generator but no cogenerator, it does not follow that C opC^{op} is representably concrete if CC is.

Characterization

Theorem

A finitely complete category is concretizable, i.e., admits a faithful functor to SetSet, if and only if it is well-powered with respect to regular subobjects.

Proof

“Only if” was proven in (Isbell). To prove it, note that if F:CDF: C\to D is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that m:axm\colon a \to x is the equalizer of f,g:xyf,g\colon x\rightrightarrows y, and n:bxn\colon b\to x is the equalizer of h,k:xzh,k\colon x\rightrightarrows z. If F(a)F(b)F(a) \cong F(b) as subobjects of F(x)F(x), then since fm=gmf m = g m and so F(f)F(m)=F(g)F(m)F(f)\circ F(m) = F(g)\circ F(m), we must also have F(f)F(n)=F(g)F(n)F(f)\circ F(n) = F(g)\circ F(n); hence (since FF is faithful) fn=gnf n = g n, so that nn factors through mm. Conversely, nn factors through mm, so we have aba\cong b as subobjects of xx. Since SetSet is regularly well-powered, it follows that any category admitting a faithful functor to SetSet must also be so.

(Actually, Isbell proved a more general condition which applies to categories that may lack finite limits.)

“If” was proven in (Freyd). The argument is rather more involved, passing through additive categories, and is not reproduced here.

Remark

A relatively deep application of Isbell’s result is that the homotopy category of topological spaces is not concretizable, even though it is a quotient of TopTop which is concretizable.

References

  • John Isbell, Two set-theoretical theorems in categories, Fund. Math 53 (1963)

Revised on December 2, 2013 11:10:26 by Todd Trimble (67.81.95.215)