For more extensive motivational background and historical notes, see Category Theory and the List Of Category Theory Topics .
A ''C'' consists of
- a Class ob(''C'') of :
- a class hom(''C'') of . Each morphism ''f'' has a unique ''source object a'' and ''target object b'' where ''a'' and ''b'' are in ob(''C''). We write ''f'': ''a'' → ''b'', and we say "''f'' is a morphism from ''a'' to ''b''". We write hom(''a'', ''b'') (or hom''C''(''a'', ''b'')) to denote the ''hom-class'' of all morphisms from ''a'' to ''b''. (Some authors write Mor(''a'', ''b'').)
- for every three objects ''a'', ''b'' and ''c'', a binary operation hom(''a'', ''b'') × hom(''b'', ''c'') → hom(''a'', ''c'') called ''composition of morphisms''; the composition of ''f'' : ''a'' → ''b'' and ''g'' : ''b'' → ''c'' is written as ''g'' o ''f'' or ''gf'' (Some authors write ''fg''.)
such that the following axioms hold:
- (associativity) if ''f'' : ''a'' → ''b'', ''g'' : ''b'' → ''c'' and ''h'' : ''c'' → ''d'' then ''h'' o (''g'' o ''f'') = (''h'' o ''g'') o ''f'', and
- (identity) for every object ''x'', there exists a morphism 1''x'' : ''x'' → ''x'' called the ''identity morphism for x'', such that for every morphism ''f'' : ''a'' → ''b'', we have 1''b'' o ''f'' = ''f'' = ''f'' o 1''a''.
From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.
A is a category in which both ob(''C'') and hom(''C'') are actually Set s and not proper classes. A category that is not small is said to be '''large'''. A '''locally small category''' is a category such that for all objects ''a'' and ''b'', the hom-class hom(''a'', ''b'') is a set. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small.
The morphisms of a category are sometimes called ''arrows'' due to the influence of Commutative Diagram s.
Each category is presented in terms of its objects, its morphisms, and its composition of morphisms.
- The category of all Set s together with Functions between sets, where composition is the usual function composition (The following are Subcategories of , obtained by adding some type of structure onto a set, by requiring that morphisms are functions that respect this added structure, and where morphism composition is simply ordinary function composition.)
- ---The category of all Preordered Sets with Monotonic Function s
- ---The category consisting of all Magmas with their Homomorphism s
- ---The category consisting of all Medial Magmas with their Homomorphism s
- ---The category consisting of all Groups with their Group Homomorphism s
- ---The category consisting of all Abelian Group s with their Group Homomorphism s
- ---The category ''K'' of all Vector Space s over the Field ''K'' (which is held fixed) with their ''K''- Linear Map s
- ---The category of all Topological Space s with Continuous functions
- ---The category of all Metric Space s with Short Map s
- ---The category of all Uniform Space s with Uniformly Continuous Function s
- The category of all small categories with functors
- Any Preordered Set (''P'', ≤) forms a small category, where the objects are the members of ''P'', the morphisms are arrows pointing from ''x'' to ''y'' when ''x'' ≤ ''y'' (The composition law is forced, because there is at most one morphism from any object to another.)
- Any Monoid forms a small category with a single object ''x''. (Here, ''x'' is any fixed set.) The morphisms from ''x'' to ''x'' are precisely the elements of the monoid, and the categorical composition of morphisms is given by the monoid operation. In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
- Any of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths. This is called the ''free category'' generated by the graph.
- If ''I'' is a Set , the '' Discrete Category on I'' is the small category that has the elements of ''I'' as objects and only the identity morphisms as morphisms. Again, the composition law is forced.)
- Any category ''C'' can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the ''dual'' Or ''opposite Category'' and is denoted ''C''op.
- If ''C'' and ''D'' are categories, one can form the ''product category'' ''C'' × ''D'': the objects are pairs consisting of one object from ''C'' and one from ''D'', and the morphisms are also pairs, consisting of one morphism in ''C'' and one in ''D''. Such pairs can be composed componentwise.
A Morphism ''f'' : ''a'' → ''b'' is called
- a '' Monomorphism '' (or ''monic'') if ''fg1'' = ''fg2'' implies ''g1'' = ''g2'' for all morphisms ''g''1, ''g2'' : ''x'' → ''a''.
- an '' Epimorphism '' (or ''epic'') if ''g1f'' = ''g2f'' implies ''g1'' = ''g2'' for all morphisms ''g1'', ''g2'' : ''b'' → ''x''.
- a if it is both a monomorphism and an epimorphism.
- a '' Retraction '' if it has a right inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''fg'' = 1''b''.
- a '' Section '' if it has a left inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''gf'' = 1''a''.
- an '' Isomorphism '' if it has an inverse, i.e. if there exists a morphism ''g'' : ''b'' → ''a'' with ''fg'' = 1''b'' and ''gf'' = 1''a''.
- an '' Endomorphism '' if ''a'' = ''b''. The class of endomorphisms of ''a'' is denoted end(''a'').
- an '' Automorphism '' if ''f'' is both an endomorphism and an isomorphism. The class of automorphisms of ''a'' is denoted aut(''a'').
Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:
- ''f'' is a monomorphism and a retraction;
- ''f'' is an epimorphism and a section;
- ''f'' is an isomorphism.
Relations among morphisms (such as ''fg'' = ''h'') can most conveniently be represented with Commutative Diagram s, where the objects are represented as points and the morphisms as arrows.
- In many categories, the hom-sets hom(''a'', ''b'') are not just sets but actually Abelian Group s, and the composition of morphisms is compatible with these group structures; i.e. is Bilinear . Such a category is called Preadditive . If, furthermore, the category has all finite Products and Coproduct s, it is called an Additive Category . If all morphisms have a Kernel and a Cokernel , and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an Abelian Category . A typical example of an abelian category is the category of abelian groups.
- A category is called Complete if all Limits exist in it. The categories of sets, abelian groups and topological spaces are complete.
- A category is called Cartesian Closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
- A Topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
- A Groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, Group Action s and Equivalence Relation s.
- Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). ''Abstract and Concrete Categories'' . Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
- Asperti, Andrea, & Longo, Giuseppe (1991). ''Categories, Types and Structures'' . Originally publ. M.I.T. Press.
- Barr, Michael, & Wells, Charles (2002). ''Toposes, Triples and Theories'' . (revised and corrected free online version of ''Grundlehren der mathematischen Wissenschaften (278).'' Springer-Verlag,1983)
- Borceux, Francis (1994). ''Handbook of Categorical Algebra.''. Vols. 50-52 of ''Encyclopedia of Mathematics and its Applications.'' Cambridge: Cambridge University Press.
- Lawvere, William, & Schanuel, Steve. (1997). ''Conceptual Mathematics: A First Introduction to Categories''. Cambridge: Cambridge University Press.
- Mac Lane, Saunders (1998). ''Categories for the Working Mathematician'' (2nd ed.). Graduate Texts in Mathematics 5. Springer. ISBN 0-387-98403-8.
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