# Grothendieck construction

The Grothendieck construction is a construction used in the mathematical field of category theory.

Let ${\displaystyle F\colon C\rightarrow \mathbf {Cat} }$ be a functor from any small category to the category of small categories. The Grothendieck construction for ${\displaystyle F}$ is the category ${\displaystyle \Gamma (F)}$ (also written ${\displaystyle C\int F}$ or ${\displaystyle F\rtimes C}$), with

• objects being pairs ${\displaystyle (c,x)}$, where ${\displaystyle c\in \operatorname {obj} (C)}$ and ${\displaystyle x\in \operatorname {obj} (F(c))}$; and
• morphisms in ${\displaystyle \operatorname {hom} _{\Gamma (F)}((c_{1},x_{1}),(c_{2},x_{2}))}$ being pairs ${\displaystyle (f,g)}$ such that ${\displaystyle f:c_{1}\to c_{2}}$ in ${\displaystyle C}$, and ${\displaystyle g:F(f)(x_{1})\to x_{2}}$ in ${\displaystyle F(c_{2})}$.

Composition of morphisms is defined by ${\displaystyle (f,g)\circ (f',g')=(f\circ f',g\circ F(f)(g'))}$.