Concrete category

Inmathematics,aconcrete categoryis acategorythat is equipped with afaithful functorto thecategory of sets(or sometimes to another category,seeRelative concretenessbelow). This functor makes it possible to think of the objects of the category as sets with additionalstructure,and of itsmorphismsas structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example thecategory of topological spacesand thecategory of groups,and trivially also the category of sets itself. On the other hand, thehomotopy category of topological spacesis notconcretizable,i.e. it does not admit a faithful functor to the category of sets.

A concrete category, when defined without reference to the notion of a category, consists of aclassofobjects,each equipped with anunderlying set;and for any two objectsAandBa set of functions, calledhomomorphisms,from the underlying set ofAto the underlying set ofB.Furthermore, for every objectA,the identity function on the underlying set ofAmust be a homomorphism fromAtoA,and the composition of a homomorphism fromAtoBfollowed by a homomorphism fromBtoCmust be a homomorphism fromAtoC.

Definition

Aconcrete categoryis a pair (C,U) such that

• Cis a category, and
• U:C→Set(the category of sets and functions) is afaithful functor.

The functorUis to be thought of as aforgetful functor,which assigns to every object ofCits "underlying set", and to every morphism inCits "underlying function".

It is customary to call the morphisms in a concrete categoryhomomorphisms(e.g., group homomorphisms, ring homomorphisms, etc.) Because of the faithfulness of the functorU,the homomorphisms of a concrete category may be formally identified with their underlying functions (i.e., their images underU); the homomorphisms then regain the usual interpretation as "structure-preserving" functions.

A categoryCisconcretizableif there exists a concrete category (C,U); i.e., if there exists a faithful functorU:C→Set.All small categories are concretizable: defineUso that its object part maps each objectbofCto the set of all morphisms ofCwhosecodomainisb(i.e. all morphisms of the formf:a→bfor any objectaofC), and its morphism part maps each morphismg:b→cofCto the functionU(g):U(b) →U(c) which maps each memberf:a→bofU(b) to the compositiongf:a→c,a member ofU(c). (Item 6 underFurther examplesexpresses the sameUin less elementary language via presheaves.) TheCounter-examplessection exhibits two large categories that are not concretizable.

Remarks

Contrary to intuition, concreteness is not apropertythat a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. In particular, a categoryCmay admit several faithful functors intoSet.Hence there may be several concrete categories (C,U) all corresponding to the same categoryC.

In practice, however, the choice of faithful functor is often clear and in this case we simply speak of the "concrete categoryC".For example," the concrete categorySet"means the pair (Set,I) whereIdenotes theidentity functorSet→Set.

The requirement thatUbe faithful means that it maps different morphisms between the same objects to different functions. However,Umay map different objects to the same set and, if this occurs, it will also map different morphisms to the same function.

For example, ifSandTare two different topologies on the same setX,then (X,S) and (X,T) are distinct objects in the categoryTopof topological spaces and continuous maps, but mapped to the same setXby the forgetful functorTop→Set.Moreover, the identity morphism (X,S) → (X,S) and the identity morphism (X,T) → (X,T) are considered distinct morphisms inTop,but they have the same underlying function, namely the identity function onX.

Similarly, any set with four elements can be given two non-isomorphic group structures: one isomorphic to$\mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z}$,and the other isomorphic to$\mathbb {Z} /4\mathbb {Z}$.

Further examples

1. Any groupGmay be regarded as an "abstract" category with one arbitrary object,$\ast$,and one morphism for each element of the group. This would not be counted as concrete according to the intuitive notion described at the top of this article. But every faithfulG-set(equivalently, every representation ofGas agroup of permutations) determines a faithful functorG→Set.Since every group acts faithfully on itself,Gcan be made into a concrete category in at least one way.
2. Similarly, anyposetPmay be regarded as an abstract category with a unique arrowx→ywheneverx≤y.This can be made concrete by defining a functorD:P→Setwhich maps each objectxto$D(x)=\{a\in P:a\leq x\}$and each arrowx→yto the inclusion map$D(x)\hookrightarrow D(y)$.
3. The categoryRelwhose objects aresetsand whose morphisms arerelationscan be made concrete by takingUto map each setXto its power set$2^{X}$and each relation$R\subseteq X\times Y$to the function$\rho:2^{X}\rightarrow 2^{Y}$defined by$\rho (A)=\{y\in Y\mid \exists \,x\in A:xRy\}$.Noting that power sets arecomplete latticesunder inclusion, those functions between them arising from some relationRin this way are exactly thesupremum-preserving maps.HenceRelis equivalent to a full subcategory of the categorySupofcomplete latticesand their sup-preserving maps. Conversely, starting from this equivalence we can recoverUas the compositeRel→Sup→Setof the forgetful functor forSupwith this embedding ofRelinSup.
4. The categorySet^opcan be embedded intoRelby representing each set as itself and each functionf:X→Yas the relation fromYtoXformed as the set of pairs (f(x),x) for allx∈X;henceSet^opis concretizable. The forgetful functor which arises in this way is thecontravariant powerset functorSet^op→Set.
5. It follows from the previous example that the opposite of any concretizable categoryCis again concretizable, since ifUis a faithful functorC→SetthenC^opmay be equipped with the compositeC^op→Set^op→Set.
6. IfCis any small category, then there exists a faithful functorP:Set^Cop→Setwhich maps a presheafXto the coproduct$\coprod _{c\in \mathrm {ob} C}X(c)$.By composing this with theYoneda embeddingY:C→Set^Copone obtains a faithful functorC→Set.
7. For technical reasons, the categoryBan₁ofBanach spacesandlinear contractionsis often equipped not with the "obvious" forgetful functor but the functorU₁:Ban₁→Setwhich maps a Banach space to its (closed)unit ball.
8. The categoryCatwhose objects are small categories and whose morphisms are functors can be made concrete by sending each categoryCto the set containing its objects and morphisms. Functors can be simply viewed as functions acting on the objects and morphisms.

Counter-examples

The categoryhTop,where the objects aretopological spacesand the morphisms arehomotopy classesof continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not existanyfaithful functor fromhToptoSetwas first proven byPeter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories andnatural equivalence-classes of functors "also fails to be concretizable.

Implicit structure of concrete categories

Given a concrete category (C,U) and acardinal numberN,letU^Nbe the functorC→Setdetermined byU^N(c) = (U(c))^N. Then asubfunctorofU^Nis called anN-ary predicateand a natural transformationU^N→UanN-ary operation.

The class of allN-ary predicates andN-ary operations of a concrete category (C,U), withNranging over the class of all cardinal numbers, forms alargesignature.The category of models for this signature then contains a full subcategory which isequivalenttoC.

Relative concreteness

In some parts of category theory, most notablytopos theory,it is common to replace the categorySetwith a different categoryX,often called abase category. For this reason, it makes sense to call a pair (C,U) whereCis a category andUa faithful functorC→Xaconcrete category overX. For example, it may be useful to think of the models of a theorywithNsortsas forming a concrete category overSet^N.

In this context, a concrete category overSetis sometimes called aconstruct.

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990).Abstract and Concrete Categories(4.2MB PDF). Originally publ. John Wiley & Sons.ISBN0-471-60922-6.(now free on-line edition).
• Freyd, Peter; (1970).Homotopy is not concrete.Originally published in: The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168. Republished in a free on-line journal: Reprints in Theory and Applications of Categories, No. 6 (2004), with the permission of Springer-Verlag.
• Rosický, Jiří; (1981).Concrete categories and infinitary languages.Journal of Pure and Applied Algebra,Volume 22, Issue 3.