comma category

English

comma category (plural comma categories)

(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 ${\\displaystyle T(g)}$ .