Grothendieck construction

If ${\displaystyle \left\{A_{i}\right\}_{i\in I}}$ is a family of sets indexed by another set, one can form the disjoint union or coproduct

${\displaystyle \coprod _{i\in I}A_{i}}$,

which is the set of all ordered pairs ${\displaystyle (i,a)}$ such that ${\displaystyle a\in A_{i}}$. The disjoint union set is naturally equipped with a "projection" map

${\displaystyle \pi$ :\coprod _{i\in I}A_{i}\to I}

defined by

${\displaystyle \pi (i,a)=i}$.

From the projection ${\displaystyle \pi }$ it is possible to reconstruct the original family of sets ${\displaystyle \left\{A_{i}\right\}_{i\in I}}$ up to a canonical bijection, as for each ${\displaystyle i\in I,A_{i}\cong \pi ^{-1}(\{i\})}$ via the bijection ${\displaystyle a\mapsto (i,a)}$. In this context, for ${\displaystyle i\in I}$, the preimage ${\displaystyle \pi ^{-1}(\{i\})}$ of the singleton set ${\displaystyle \{i\}}$ is called the "fiber" of ${\displaystyle \pi }$ over ${\displaystyle i}$, and any set ${\displaystyle B}$ equipped with a choice of function ${\displaystyle f:B\to I}$ is said to be "fibered" over ${\displaystyle I}$. In this way, the disjoint union construction provides a way of viewing any of sets indexed by ${\displaystyle I}$ as a set "fibered" over ${\displaystyle I}$, and conversely, for any set ${\displaystyle f:B\to I}$ fibered over ${\displaystyle I}$, we can view it as the disjoint union of the fibers of ${\displaystyle f}$. Jacobs has referred to this these two perspectives as "display indexing" and "pointwise indexing", see.[2]

The Grothendieck construction generalizes this to categories. For each category ${\displaystyle {\mathcal {C}}}$, family of categories ${\displaystyle \{F(c)\}_{c\in {\mathcal {C}}}}$ indexed by the objects of ${\displaystyle {\mathcal {C}}}$ in a functorial way, the Grothendieck construction returns a new category ${\displaystyle {\mathcal {E}}}$ fibered over ${\displaystyle {\mathcal {C}}}$ by a functor ${\displaystyle \pi }$ whose fibers are the categories ${\displaystyle \{F(c)\}_{c\in {\mathcal {C}}}}$.

Let ${\displaystyle F\colon {\mathcal {C}}\rightarrow \mathbf {Cat} }$ be a functor from any small category to the category of small categories. The Grothendieck construction for ${\displaystyle F}$ is the category ${\displaystyle \Gamma (F)}$ (also written ${\displaystyle \textstyle \int _{\textstyle {\mathcal {C}}}F}$, ${\displaystyle \textstyle {\mathcal {C}}\int F}$ or ${\displaystyle F\rtimes {\mathcal {C}}}$), with

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

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

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

• Category of elements

• 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.

Specific

• Grothendieck Construction at the nLab