category of small categories

Notation

\Cat

Objects small categories Morphisms functors Related

\Mon

\Set

\PreOrd

\DiGraph

External nLab Link

This is the category of small categories and functors between them. It is the prototype of a 2-category, but here we only treat it as a 1-category.

Satisfied Properties

Assigned properties

Deduced properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

terminal object: trivial category
initial object: empty category
products: direct products with pointwise operations
coproducts: disjoint unions with no morphisms between objects in distinct summands

isomorphisms: functors that are bijective on objects and morphisms
monomorphisms: faithful functors that are injective on objects
epimorphisms: A functor $F : \C \to \D$ is an epimorphism iff $F$ is surjective on objects and for every morphism $s$ in $\D$ there is a zigzag over $U \coloneqq F(\C)$ , meaning morphisms $u_1,\dotsc,u_{m+1} \in U$ , $v_1,\dotsc,v_m \in U$ , $x_1,\dotsc,x_m \in \D$ and $y_1,\dotsc,y_m \in \D$ such that $s = x_1 u_1$ , $u_1 = v_1 y_1$ , $x_{i-1} v_{i-1} = x_i u_i$ , $u_i y_{i-1} = v_i y_i$ , $x_m v_m = u_{m+1}$ and $u_{m+1} y_m = s$ .
regular monomorphisms:
regular epimorphisms:

The database stores 2 functors based on the category of small categories.

The database stores 1 morphism based on the category of small categories.

The database stores 1 symmetric monoidal category based on the category of small categories.