Grothendieck construction

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

Let $F\colon C\rightarrow \mathbf {Cat}$ 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 $C\int F$ or $F\rtimes C$ ), with

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

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

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Grothendieck construction

See also

References

External links