concrete category

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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a concrete category is a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
that is equipped with a
faithful functorIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
to the
category of setsIn the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...
(or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of the category as sets with additional
structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A ...
, and of its
morphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the
category of topological spacesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and the
category of groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and trivially also the category of sets itself. On the other hand, the
homotopy category of topological spacesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is not concretizable, 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 a class (set theory), class of ''objects'', each equipped with an ''underlying set''; and for any two objects ''A'' and ''B'' a set of functions, called ''morphisms'', from the underlying set of ''A'' to the underlying set of ''B''. Furthermore, for every object ''A'', the identity function on the underlying set of ''A'' must be a morphism from ''A'' to ''A'', and the composition of a morphism from ''A'' to ''B'' followed by a morphism from ''B'' to ''C'' must be a morphism from ''A'' to ''C''.

# Definition

A concrete category is a pair (''C'',''U'') such that *''C'' is a category, and *''U'' : ''C'' → Set (the category of sets and functions) is a
faithful functorIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
. 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 ''U'': ''C'' → Set. All small categories are concretizable: define ''U'' so that its object part maps each object ''b'' of ''C'' to the set of all morphisms of ''C'' whose codomain is ''b'' (i.e. all morphisms of the form ''f'': ''a'' → ''b'' for any object ''a'' of ''C''), and its morphism part maps each morphism ''g'': ''b'' → ''c'' of ''C'' to the function ''U''(''g''): ''U''(''b'') → ''U''(''c'') which maps each member ''f'': ''a'' → ''b'' of ''U''(''b'') to the composition ''gf'': ''a'' → ''c'', a member of ''U''(''c''). (Item 6 under concrete category#Further examples, Further examples expresses the same ''U'' in less elementary language via presheaves.) The concrete category#Counter-examples, Counter-examples section exhibits two large categories that are not concretizable.

# Remarks

It is important to note that, contrary to intuition, concreteness is not a property (philosophy), 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 faithful 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. 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 the category Top of topological spaces and continuous maps, but mapped to the same set ''X'' by the forgetful functor Top → Set. 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 four elements can be given two non-isomorphic group structures: one isomorphic to $\mathbb/2\mathbb \times \mathbb/2\mathbb$, and the other isomorphic to $\mathbb/4\mathbb$.

# Further examples

# Any group ''G'' may 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 faithful Group action (mathematics), ''G''-set (equivalently, every representation of ''G'' as a permutation group, 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. # 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\left(x\right)=\$ and each arrow ''x'' → ''y'' to the inclusion map $D\left(x\right) \hookrightarrow D\left(y\right)$. # The category Category of relations, Rel whose objects are Set (mathematics), sets and whose morphisms are Relation (mathematics), relations can be made concrete by taking ''U'' to map each set ''X'' to 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\left(A\right)=\$. Noting that power sets are complete lattices under inclusion, those functions between them arising from some relation ''R'' in this way are exactly the Complete lattice#Morphisms of complete lattices, supremum-preserving maps. Hence Rel is equivalent to a full subcategory of the category Sup of complete lattices and their sup-preserving maps. Conversely, starting from this equivalence we can recover ''U'' as the composite Rel → Sup → Set of the forgetful functor for Sup with this embedding of Rel in Sup. # The category Setop can be embedded into Rel by representing each set as itself and each function ''f'': ''X'' → ''Y'' as the relation from ''Y'' to ''X'' formed as the set of pairs (''f''(''x''), ''x'') for all ''x'' ∈ ''X''; hence Setop is concretizable. The forgetful functor which arises in this way is the Functor#Examples, contravariant powerset functor Setop → Set. # 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 ''C''op may be equipped with the composite ''C''op → Setop → Set. # If ''C'' is any small category, then there exists a faithful functor ''P'' : Set''C''op → Set which maps a presheaf ''X'' to the coproduct $\coprod_ X\left(c\right)$. By composing this with the Yoneda embedding ''Y'':''C'' → Set''C''op one obtains a faithful functor ''C'' → Set. # For technical reasons, the category Ban1 of Banach spaces and contraction (operator theory), linear contractions is often equipped not with the "obvious" forgetful functor but the functor ''U''1 : Ban1 → Set which maps a Banach space to its (closed) unit ball. #The category Cat whose objects are small categories and whose morphisms are functors can be made concrete by sending each category C to the set containing its objects and morphisms. Functors can be simply viewed as functions acting on the objects and morphisms.

# Counter-examples

The category homotopy category of topological spaces, hTop, where the objects are topological spaces and the morphisms are homotopy, 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.

# 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 proper class, large signature (logic), signature. The category of models for this signature then contains a full subcategory which is equivalence of categories, equivalent to ''C''.

# 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 Structure (mathematical logic)#Many-sorted structures, with ''N'' sorts as forming a concrete category over Set''N''. In this context, a concrete category over Set is sometimes called a ''construct''.

# References

* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990)
''Abstract and Concrete Categories''
(4.2MB PDF). Originally publ. John Wiley & Sons. {{ISBN, 0-471-60922-6. (now free on-line edition). * Freyd, Peter; (1970)

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. Category theory