Types and Examples of Categories |

**General Types**

* __Single spaces as categories__:
Any set with a relation is a category, with relations as arrows; For example, a poset.

* __Thin__: One for which each Hom-set
contains at most one element; For example, a poset.

* __With terminal element__: One with
an object to which there is an arrow from every other object; For example, Set.

* __Tensor categories__: Types are spherical,
ribbon, symmetric; > s.a. lattice gauge theory.

* __Remark__: Some categories are
modeled after other ones; For example, manifolds are modeled after finite-dimensional
vector spaces.

@ __Other types__:
Neeman Top(98) [non-compactly generated];
Jacobs a1101 [dagger categories of tame relations].

> __Online resources__:
see Wikipedia page.

**Examples**
s.a. category theory [generalizations]; categories
in physics [categories of relations, applications].

* __Set__: The category
of sets and mappings between sets.

* __Top__: The category
of topological spaces, with continuous maps as morphisms; The
composition is composition of maps.

* __Grp__: The category
of groups, with homomorphisms as morphisms; Composition is composition
of maps; A monomorphism is a 1-1 homomorphism *f*: Ker(*f*)
= *e* (the identity).

* __Hilb__: The category of
Hilbert spaces and bounded linear operators, used in quantum mechanics.

* __Man__: The category of
differentiable manifolds and differentiable maps.

* __nCob__: The category
of manifolds (= spacelike hypersurfaces) and *n*-dimensional
cobordisms, used in general relativity.

* __Prop__: A strict symmetric
monoidal category where the objects are natural numbers, with the tensor
product of objects given by addition.

* __Vec__: The category of
vector spaces, a monoidal tensor category.

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send feedback and suggestions to bombelli at olemiss.edu – modified 24 dec 2017