Concrete category

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In mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose morphisms are structure-preserving functions, and whose composition operation is composition of functions. The formal definition does not coincide perfectly with this intuition.

The category of sets and functions Set is trivially a concrete category, since every set can be thought of as carrying a trivial structure. Further important examples include Top, the category of topological spaces and continuous functions, and Grp the category of groups and group homomorphisms.

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[edit] Definition

A concrete category is a pair (C,U) such that

The functor U is to be thought of as a forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C its "underlying function".

A category C is concretizable if there exists a concrete category (C,U); i.e., if there exists a faithful functor from U:C → Set.

[edit] Remarks

  1. It is important to note that, contrary to intuition, concreteness is not a property which a category may or may not satisfy, but rather a structure with which a category may or may not be equipped. In particular, a category C may admit several faithful functors into Set. Hence there may be several concrete categories (C,U) all corresponding to the same category C. In practice, however, the choice of forgetful functor is often clear and in this case we simply speak of the "concrete category C". For example, "the concrete category Set" means the pair (Set,I) where I denotes the identity functor Set → Set.
  2. The requirement that U be faithful means that it maps different morphisms between the same objects to different functions. However, U may map different objects to the same set and, if this occurs, it will also map different morphisms to the same function. For example, if S and T are two different topologies on the same set X, then (X,S) and (X,T) are distinct objects in Top which the forgetful functor Top → Set maps to the same set, namely X. Moreover, the identity morphism (X,S) → (X,S) and the identity morphism (X,T) → (X,T) are considered distinct morphisms in Top, but they have the same underlying function, namely the identity function on X. Similarly, any set with 4 elements can be given two non-isomorphic group structures: one isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}; the other isomorphic to \mathbb{Z}/4\mathbb{Z}).

[edit] Further examples

  1. Any group G may be regarded as an "abstract" category with one 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 faithful G-set (equivalently, every representation of G as a group of permutations) determines a faithful functor G → Set. Since every group acts faithfully on itself, G can be made into a concrete category in at least one way.
  2. Similarly, any poset P may be regarded as an abstract category with a unique arrow x → y whenever x ≤ y. This can be made concrete by defining a functor D : P → Set which maps each object x to D(x)=\{a \in P : a \leq x\} and each arrow x → y to the inclusion map D(x) \hookrightarrow D(y).
  3. The category Rel whose objects are sets and whose morphisms are relations may not appear, at first glance, to be concretizable. It is, however, equivalent to a full subcategory of the category Sup whose objects are complete lattices and whose morphisms are supremum-preserving maps. The latter is concrete, so we can equip Rel with the composite Rel → Sup → Set. If we do so, then the "underlying set" of an object of Rel (i.e., a set) is not itself, but rather its powerset. The "underlying function" of a relation R \subseteq X \times Y in this sense is the function \rho: 2^X \rightarrow 2^Y defined by \rho(A)=\{y \in Y : \exists x \in A ((x,y)\in R)\}.
  4. The category Setop can be embedded into Rel; hence it too is concretizable. The forgetful functor which arises in this way is the contravariant powerset functor Setop → Set.
  5. It follows from the previous example that the opposite of any concretizable category C is again concretizable, since if U is a faithful functor C → Set then Cop may be equipped with the composite Cop → Setop → Set.
  6. If C is any small category, then there exists a faithful functor P : SetCop → Set which maps a presheaf X to the product \prod_{c \in \mathrm{ob}C} X(c). By composing this with the Yoneda embedding Y:C → SetCop one obtains a faithful functor C → Set.
  7. For technical reasons, the category Ban1 of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor U1 : Ban1 → Set which maps a Banach space to its (closed) unit ball.

[edit] Counter-examples

The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of 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 exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable.

[edit] Implicit structure of concrete categories

Given a concrete category (C,U) and a cardinal number N, let UN be the functor C → Set determined by UN(c) = (U(c))N. Then a subfunctor of UN is called an N-ary predicate and a natural transformation UN → U an N-ary operation.

The class of all N-ary predicates and N-ary operations of a concrete category (C,U), with N ranging over the class of all cardinal numbers, forms a large signature. The category of models for this signature then contains a full subcategory which is equivalent to C.

[edit] Relative concreteness

In some parts of category theory, most notably topos theory, it is common to replace the category Set with a different category X, often called a base category. For this reason, it makes sense to call a pair (C,U) where C is a category and U a faithful functor C → X a concrete category over X. For example, it may be useful to think of the models of a theory with N sorts as forming a concrete category over SetN.

In this context, a concrete category over Set is sometimes called a construct.

[edit] References

  • 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.
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