Grothendieck construction

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

Let

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

be a functor from any small category to the category of small categories. The Grothendieck construction for $F$ is the category $\Gamma(F)$ (also written $C \int F$), with

• objects being pairs $(c,x)$, where $c\in obj(C)$ and $x\in obj(F(c))$; and
• morphisms $\text{Hom}_{\Gamma(F)}((c_1,x_1),(c_2,x_2))$ being pairs $(f, x)$ such that $f\colon c_1\rightarrow c_2$ in $mor(C)$ and $x\colon F(f)(x_1) \rightarrow x_2$ in $mor(F(c_2))$.

Composition of morphisms is defined by $(f, x) \cdot (f', x') = (f f', x \cdot F(f) (x'))$.

See Also

• Category of elements

References

• Mac Lane and Moerdijk, Sheaves in Geometry and Logic, pp. 44.
• R. W. Thomason (1979). Homotopy colimits in the category of small categories. Mathematical Proceedings of the Cambridge Philosophical Society, 85, pp 91-109. doi:10.1017/S0305004100055535.

External links

• Grothendieck Construction in nLab