Editing Equivalence of categories (section)

==Examples==
* Consider the category <math>C</math> having a single object <math>c</math> and a single morphism <math>1_{c}</math>, and the category <math>D</math> with two objects <math>d_{1}</math>, <math>d_{2}</math> and four morphisms: two identity morphisms <math>1_{d_{1}}</math>, <math>1_{d_{2}}</math> and two isomorphisms <math>\alpha \colon d_{1} \to d_{2}</math> and <math>\beta \colon d_{2} \to d_{1}</math>.  The categories <math>C</math> and <math>D</math> are equivalent; we can (for example) have <math>F</math> map <math>c</math> to <math>d_{1}</math> and <math>G</math> map both objects of <math>D</math> to <math>c</math> and all morphisms to <math>1_{c}</math>.
* By contrast, the category <math>C</math> with a single object and a single morphism is ''not'' equivalent to the category <math>E</math> with two objects and only two identity morphisms. The two objects in <math>E</math>  are ''not'' isomorphic in that there are no morphisms between them. Thus any functor from <math>C</math> to <math>E</math> will not be essentially surjective.
* Consider a category <math>C</math> with one object <math>c</math>, and two morphisms <math>1_{c}, f \colon c \to c</math>.  Let <math>1_{c}</math> be the identity morphism on <math>c</math> and set <math>f \circ f = 1</math>.  Of course, <math>C</math> is equivalent to itself, which can be shown by taking <math>1_{c}</math> in place of the required natural isomorphisms between the functor <math>\mathbf{I}_{C}</math> and itself.  However, it is also true that <math>f</math> yields a natural isomorphism from <math>\mathbf{I}_{C}</math> to itself.  Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.
* The [[category of sets]] and [[partial function]]s is equivalent to but not isomorphic with the category of [[pointed set]]s and point-preserving maps.<ref name="KoslowskiMelton2001">{{cite book|editor=Jürgen Koslowski and Austin Melton|title=Categorical Perspectives|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4186-3|page=10|author=Lutz Schröder|chapter=Categories: a free tour}}</ref>
* Consider the category <math>C</math> of finite-[[dimension of a vector space|dimensional]] [[real number|real]] [[vector space]]s, and the category <math>D = \mathrm{Mat}(\mathbb{R})</math> of all real [[matrix (mathematics)|matrices]] (the latter category is explained in the article on [[additive category|additive categories]]).  Then <math>C</math> and <math>D</math> are equivalent: The functor <math>G \colon D \to C</math> which maps the object <math>A_{n}</math> of <math>D</math> to the vector space <math>\mathbb{R}^{n}</math> and the matrices in <math>D</math> to the corresponding linear maps is full, faithful and essentially surjective.
* One of the central themes of [[algebraic geometry]] is the duality of the category of [[affine scheme]]s and the category of [[commutative ring]]s.  The functor <math>G</math> associates to every commutative ring its [[spectrum of a ring|spectrum]], the scheme defined by the [[prime ideal]]s of the ring.  Its adjoint <math>F</math> associates to every affine scheme its ring of global sections.
* In [[functional analysis]] the category of commutative [[C*-algebra]]s with identity is contravariantly equivalent to the category of [[compact space|compact]] [[Hausdorff space]]s.  Under this duality, every compact Hausdorff space <math>X</math> is associated with the algebra of continuous complex-valued functions on <math>X</math>, and every commutative C*-algebra is associated with the space of its [[maximal ideal]]s.  This is the [[Gelfand representation]].
* In [[lattice theory]], there are a number of dualities, based on [[representation theorem]]s that connect certain classes of lattices to classes of [[topology|topological spaces]].  Probably the most well-known theorem of this kind is ''[[Stone's representation theorem for Boolean algebras]]'', which is a special instance within the general scheme of ''[[Stone duality]]''.  Each [[Boolean algebra (structure)|Boolean algebra]] <math>B</math> is mapped to a specific topology on the set of [[lattice theory|ultrafilters]] of <math>B</math>.  Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra.  One obtains a duality between the category of Boolean algebras (with their homomorphisms) and [[Stone space]]s (with continuous mappings). Another case of Stone duality is [[Birkhoff's representation theorem]] stating a duality between finite partial orders and finite distributive lattices.
* In [[pointless topology]] the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.
* For two [[Ring (mathematics)|rings]] ''R'' and ''S'', the [[product category]] ''R''-'''Mod'''×''S''-'''Mod''' is equivalent to (''R''×''S'')-'''Mod'''.{{Citation needed|date=May 2015}}
* Any category is equivalent to its [[skeleton (category theory)|skeleton]].