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 $(f,x)\cdot (f',x')=(ff',x\cdot F(f)(x'))$ .

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

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Grothendieck construction

See also

References

External links