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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 foundational work on algebraic topology. Nowadays, cat ...
, two categories ''C'' and ''D'' are isomorphic if there exist
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
s ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' which are mutually inverse to each other, i.e. ''FG'' = 1''D'' (the identity functor on ''D'') and ''GF'' = 1''C''. This means that both the
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
s and the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s of ''C'' and ''D'' stand in a one-to-one correspondence to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms. Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
; roughly speaking, for an equivalence of categories we don't require that FG be ''equal'' to 1_D, but only ''
naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
'' to 1_D, and likewise that GF be naturally isomorphic to 1_C.


Properties

As is true for any notion of
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, we have the following general properties formally similar to an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
: * any category ''C'' is isomorphic to itself * if ''C'' is isomorphic to ''D'', then ''D'' is isomorphic to ''C'' * if ''C'' is isomorphic to ''D'' and ''D'' is isomorphic to ''E'', then ''C'' is isomorphic to ''E''. A functor ''F'' : ''C'' → ''D'' yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets. This criterion can be convenient as it avoids the need to construct the inverse functor ''G''.


Examples

* Consider a finite group ''G'', a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''k'' and the group algebra ''kG''. The category of ''k''-linear
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s of ''G'' is isomorphic to the category of
left module In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the m ...
s over ''kG''. The isomorphism can be described as follows: given a group representation ρ : ''G'' → GL(''V''), where ''V'' is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
over ''k'', GL(''V'') is the group of its ''k''-linear
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s, and ρ is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
, we turn ''V'' into a left ''kG'' module by defining \left(\sum_ a_g g\right) v = \sum_ a_g \rho(g)(v) for every ''v'' in ''V'' and every element Σ ''ag'' ''g'' in ''kG''. Conversely, given a left ''kG'' module ''M'', then ''M'' is a ''k'' vector space, and multiplication with an element ''g'' of ''G'' yields a ''k''-linear automorphism of ''M'' (since ''g'' is invertible in ''kG''), which describes a group homomorphism ''G'' → GL(''M''). (There are still several things to check: both these assignments are functors, i.e. they can be applied to maps between group representations resp. ''kG'' modules, and they are inverse to each other, both on objects and on morphisms). See also . * Every ring can be viewed as a preadditive category with a single object. The
functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in ...
of all
additive functor In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom- ...
s from this category to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is ...
is isomorphic to the category of left modules over the ring. * Another isomorphism of categories arises in the theory of
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s: the category of Boolean algebras is isomorphic to the category of
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
s. Given a Boolean algebra ''B'', we turn ''B'' into a Boolean ring by using the
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
as addition and the meet operation \land as multiplication. Conversely, given a Boolean ring ''R'', we define the join operation by ''a''\lor''b'' = ''a'' + ''b'' + ''ab'', and the meet operation as multiplication. Again, both of these assignments can be extended to morphisms to yield functors, and these functors are inverse to each other. * If ''C'' is a category with an initial object s, then the slice category (''s''↓''C'') is isomorphic to ''C''. Dually, if ''t'' is a terminal object in ''C'', the functor category (''C''↓''t'') is isomorphic to ''C''. Similarly, if 1 is the category with one object and only its identity morphism (in fact, 1 is the terminal category), and ''C'' is any category, then the functor category ''C''1, with objects functors ''c'': 1 → ''C'', selecting an object ''c''∈Ob(''C''), and arrows natural transformations ''f'': ''c'' → ''d'' between these functors, selecting a morphism ''f'': ''c'' → ''d'' in ''C'', is again isomorphic to ''C''.


See also

*
Equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...


References

{{DEFAULTSORT:Isomorphism Of Categories Adjoint functors Equivalence (mathematics)