comma categoryとは 意味・読み方・使い方

名詞

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

下位語

• arrow category
• slice category
• coslice category