Other concepts Category theory

1 other concepts

1.1 universal constructions, limits, , colimits
1.2 equivalent categories
1.3 further concepts , results
1.4 higher-dimensional categories

other concepts
universal constructions, limits, , colimits

using language of category theory, many areas of mathematical study can categorized. categories include sets, groups , topologies.

each category distinguished properties objects have in common, such empty set or product of 2 topologies, yet in definition of category, objects considered atomic, i.e., not know whether object set, topology, or other abstract concept. hence, challenge define special objects without referring internal structure of objects. define empty set without referring elements, or product topology without referring open sets, 1 can characterize these objects in terms of relations other objects, given morphisms of respective categories. thus, task find universal properties uniquely determine objects of interest.

indeed, turns out numerous important constructions can described in purely categorical way. central concept needed purpose called categorical limit, , can dualized yield notion of colimit.

equivalent categories

it natural question ask: under conditions can 2 categories considered same, in sense theorems 1 category can readily transformed theorems other category? major tool 1 employs describe such situation called equivalence of categories, given appropriate functors between 2 categories. categorical equivalence has found numerous applications in mathematics.

further concepts , results

the definitions of categories , functors provide basics of categorical algebra; additional important topics listed below. although there strong interrelations between of these topics, given order can considered guideline further reading.

the functor category d has objects functors c d , morphisms natural transformations of such functors. yoneda lemma 1 of famous basic results of category theory; describes representable functors in functor categories.
duality: every statement, theorem, or definition in category theory has dual obtained reversing arrows . if 1 statement true in category c dual true in dual category c. duality, transparent @ level of category theory, obscured in applications , can lead surprising relationships.
adjoint functors: functor can left (or right) adjoint functor maps in opposite direction. such pair of adjoint functors typically arises construction defined universal property; can seen more abstract , powerful view on universal properties.

higher-dimensional categories

many of above concepts, equivalence of categories, adjoint functor pairs, , functor categories, can situated context of higher-dimensional categories. briefly, if consider morphism between 2 objects process taking 1 object , higher-dimensional categories allow profitably generalize considering higher-dimensional processes .

for example, (strict) 2-category category morphisms between morphisms , i.e., processes allow transform 1 morphism another. can compose these bimorphisms both horizontally , vertically, , require 2-dimensional exchange law hold, relating 2 composition laws. in context, standard example cat, 2-category of (small) categories, , in example, bimorphisms of morphisms natural transformations of morphisms in usual sense. basic example consider 2-category single object; these monoidal categories. bicategories weaker notion of 2-dimensional categories in composition of morphisms not strictly associative, associative isomorphism.

this process can extended natural numbers n, , these called n-categories. there notion of ω-category corresponding ordinal number ω.

higher-dimensional categories part of broader mathematical field of higher-dimensional algebra, concept introduced ronald brown. conversational introduction these ideas, see john baez, tale of n-categories (1996).