Idea
- a collection of things and binary relationships between them, such that these relationships can be combined and include the “identity” relationship “is the same as.”
- a quiver with a rule saying how to compose two edges that fit together to get a new edge.
- furthermore, each vertex has has edge going to itself, acting as an identify for this composition.
- a combinatorial model for a directed space: …
Definitions
- : exists two broad ways to define
- one definition generalizes well to the notion of internal category, while the other generalizes well to the notion of enriched category.
- good to know both
- they are equivalent
- definition with one collection of morphisms
- a collection of objects
- a collection of morphisms
- source / domain and target / codomain: for every morphism , an object s(f) and an object t(f)
- composite: for every pair of morphisms and , where t(f) = s(g), a morphism
- identity morphism: for every object , a morphism
- such that the following properties:
- source and target are respected by composition
- source and target are respected by identity
- composition is associative
- composition satisfies the left and right unit laws
- if s(f) = x and t(f) = y, then
- semicategory: if the identity-assigning map and its axiom is omitted
- definition with a family of collections of morphisms
- a collection of objects
- for each pair , of objects, a collection of morphisms from to
- composite: for each pair of morphisms in and in , a morphism in
- identity morphisms: for every object , a morphism in
- such that the following properties hold:
- composition is associative
- composition satisfies the left and right unit laws
Alternative Definition
- single-sorted definition: a variant of the first definition
- type-theoretic definition: a variant of the second definition
- protocategory: a mixture of the first and second definitions
Equivalent Definitions
- a monad in the 2-category of spans of sets
- a monoid in the monoidal category of endospans on the set of objects
- a simplicial set which satisfies the Segal conditions
- a simplicial set which satisfies the weak Kan complex conditions strictly
Generalizations:
- Internal categories: define a category internal to some other category
- Enriched categories: define a category enriched over some other category
- Indexed categories: captures the idea of working “over a base” other than Set
- Multicategories: allows morphisms to go from several objects to a single object
- aka operad
Examples: …