Types and Examples of Categories| 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.
* R-modules:
A monomorphism μ: A → B is essential
if for any submodule H of B, if H ≠ 0 then
H ∩ μA ≠ 0.
* Non-trivial category:
@ see Spanier 66 on algebraic topology;
The composition is not the usual composition of maps.
@ Other examples: Mitchener PLMS(02) [C* categories];
Baez et al a1707 [props, and network theory].
> Other examples: see posets
[as (0,1)-categories or thin categories]; fiber bundles [natural bundles].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 24 dec 2017