• Consider the category C having a single object c and a single morphism 1_{c}, and the category D with two objects d_{1}, d_{2} and four morphisms: two identity morphisms 1_{d_{1}}, 1_{d_{2}} and two isomorphisms \alpha \colon d_{1} \to d_{2} and \beta \colon d_{2} \to d_{1}. The categories C and D are equivalent; we can (for example) have F map c to d_{1} and G map both objects of D to c and all morphisms to 1_{c}. • By contrast, the category C with a single object and a single morphism is
not equivalent to the category E with two objects and only two identity morphisms. The two objects in E are
not isomorphic in that there are no morphisms between them. Thus any functor from C to E will not be essentially surjective. • Consider a category C with one object c, and two morphisms 1_{c}, f \colon c \to c. Let 1_{c} be the identity morphism on c and set f \circ f = 1. Of course, C is equivalent to itself, which can be shown by taking 1_{c} in place of the required natural isomorphisms between the functor \mathbf{I}_{C} and itself. However, it is also true that f yields a natural isomorphism from \mathbf{I}_{C} 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 functions is equivalent to but not isomorphic with the category of
pointed sets and point-preserving maps. • Consider the category C of finite-
dimensional real vector spaces, and the category D = \mathrm{Mat}(\mathbb{R}) of all real
matrices (the latter category is explained in the article on
additive categories). Then C and D are equivalent: The functor G \colon D \to C which maps the object A_{n} of D to the vector space \mathbb{R}^{n} and the matrices in D 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 schemes and the category of
commutative rings. The functor G associates to every commutative ring its
spectrum, the scheme defined by the
prime ideals of the ring. Its adjoint F associates to every affine scheme its ring of global sections. • In
functional analysis the category of commutative
C*-algebras with identity is contravariantly equivalent to the category of
compact Hausdorff spaces. Under this duality, every compact Hausdorff space X is associated with the algebra of continuous complex-valued functions on X, and every commutative C*-algebra is associated with the space of its
maximal ideals. This is the
Gelfand representation. • In
lattice theory, there are a number of dualities, based on
representation theorems that connect certain classes of lattices to classes of
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 B is mapped to a specific topology on the set of
ultrafilters of B. 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 spaces (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
rings R and
S, the
product category R-
Mod×
S-
Mod is equivalent to (
R×
S)-
Mod. • Any category is equivalent to its
skeleton. ==Properties==