comma category

comma category (plural comma categories)

1. (category theory) A category built out of a pair of functors that have the same codomain.

   Given a pair of functors ${\displaystyle S:{\mathcal {A}}\rightarrow {\mathcal {C}}}$ and ${\displaystyle T:{\mathcal {B}}\rightarrow {\mathcal {C}}}$, objects of the comma category ${\displaystyle S\downarrow T}$ are arrows ${\displaystyle h:S(A)\rightarrow T(B)}$ parametrized by triples (A, B, h), and given morphisms ${\displaystyle f:A\rightarrow A'}$ and ${\displaystyle g:B\rightarrow B'}$, then a morphism of the said comma category is a commuting square parametrized by the pair (f, g) and spanning the area from h to ${\displaystyle h':S(A')\rightarrow T(B')}$ and from ${\displaystyle S(f)}$ to ${\displaystyle T(g)}$.

• arrow category
• slice category
• coslice category