Grothendieck construction

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

Let

$F\colon C\rightarrow \text{Sets}$

be a functor from any small category to the category of sets. The Grothendieck construct for $F$ is the category $\Gamma(F)$ whose objects are pairs $(c,x)$ , where $c\in C$ is an object and $x\in F(c)$ is an element, and for which the set $\text{Hom}_{\Gamma(F)}((c_1,x_1),(c_2,x_2))$ is the set of morphisms $f\colon c_1\rightarrow c_2$ in $C$ such that $F(f)(x_1)=x_2$ .

References

Mac Lane and Moerdijk, Sheaves in Geometry and Logic, pp. 444.