Grothendieck construction

The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory.

Definition

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

• objects being pairs $(c,x)$ , where $c\in \operatorname {obj} ({\mathcal {C}})$ 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 ${\mathcal {C}}$ , and $g:F(f)(x_{1})\to x_{2}$ in $F(c_{2})$ .

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

Slogan

"The Grothendieck construction takes structured, tabulated data and flattens it by throwing it all into one big space. The projection functor is then tasked with remembering which box each datum originally came from." 

Example

If $G$ is a group, then it can be viewed as a category, ${\mathcal {C}}_{G},$ with one object and all morphisms invertible. Let $F:{\mathcal {C}}_{G}\to \mathbf {Cat}$ be a functor whose value at the sole object of ${\mathcal {C}}_{G}$ is the category ${\mathcal {C}}_{H},$ a category representing the group $H$ in the same way. The requirement that $F$ be a functor is then equivalent to specifying a group homomorphism $\varphi :G\to \operatorname {Aut} (H),$ where $\operatorname {Aut} (H)$ denotes the group of automorphisms of $H.$ Finally, the Grothendieck construction, $F\rtimes {\mathcal {C}}_{G},$ results in a category with one object, which can again be viewed as a group, and in this case, the resulting group is (isomorphic to) the semidirect product $H\rtimes _{\varphi }G.$ 