In

^{op} 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 Set^{op} is concretizable. The forgetful functor which arises in this way is the contravariant powerset functor Set^{op} → 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} → Set^{op} → 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 $\backslash coprod\_\; X(c)$. By composing this with the Yoneda embedding ''Y'':''C'' → Set^{''C''op} one obtains a faithful functor ''C'' → Set.
# For technical reasons, the category Ban_{1} of Banach spaces and linear contractions is often equipped not with the "obvious" forgetful functor but the functor ''U''_{1} : Ban_{1} → 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.

^{N}'' be the functor ''C'' → Set determined by ''U^{N}(c) = (U(c))^{N}''.
Then a subfunctor of ''U^{N}'' is called an ''N-ary predicate'' and a
natural transformation ''U^{N}'' → ''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

^{''N''}.
In this context, a concrete category over Set is sometimes called a ''construct''.

''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

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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), ''Categories'' (Aristotle)
*Category (Kant)
...

that is equipped with a faithful functor to the category of sets
In the mathematical field of category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundatio ...

(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, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

, and of its morphism
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

s as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces 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 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 functor. 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'' whosecodomain
In mathematics, the codomain or set of destination of a Function (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...

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 Further examples expresses the same ''U'' in less elementary language via presheaves.) The Counter-examples section exhibits two large categories that are not concretizable.
Remarks

It is important to note that, contrary to intuition, concreteness is not aproperty
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to Consumables, consume, alte ...

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 $\backslash mathbb/2\backslash mathbb\; \backslash times\; \backslash mathbb/2\backslash mathbb$, and the other isomorphic to $\backslash mathbb/4\backslash mathbb$.
Further examples

# Any group ''G'' may be regarded as an "abstract" category with one arbitrary object, $\backslash ast$, 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. # 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)=\backslash $ and each arrow ''x'' → ''y'' to the inclusion map $D(x)\; \backslash hookrightarrow\; D(y)$. # The category Rel whose objects are sets and whose morphisms are relations can be made concrete by taking ''U'' to map each set ''X'' to its power set $2^X$ and each relation $R\; \backslash subseteq\; X\; \backslash times\; Y$ to the function $\backslash rho:\; 2^X\; \backslash rightarrow\; 2^Y$ defined by $\backslash rho(A)=\backslash $. Noting that power sets are complete lattices under inclusion, those functions between them arising from some relation ''R'' in this way are exactly the 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 SetCounter-examples

The categoryhTop
htop is an interactive system monitor, system-monitor process-viewer and process-manager. It is designed as an alternative to the Unix program top (Unix), top.
System monitor
It shows a frequently updated list of the processes running on a com ...

, 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.
Implicit structure of concrete categories

Given a concrete category (''C'', ''U'') and acardinal number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

''N'', let ''Usignature
A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...

. The category of models for this signature then contains a full subcategory which is 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 with ''N'' sorts as forming a concrete category over SetNotes

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