---
tag: 272T16NWZ9
kind: concept
name: Grothendieck construction
msc: [18D30, 18A25]
sources:
  - url: https://en.wikipedia.org/wiki/Category_of_elements
---

## Summary

A [[functor|28NQ0KRHD3]] $F \colon \mathcal{C} \to \mathbf{Cat}$ assigns a category to each object of $\mathcal{C}$.
The Grothendieck construction turns $F$ into a single [[category|A2DE66937D]] $\int F$ with a projection functor to $\mathcal{C}$.
It glues the categories $F(c)$ together along the functors $F(f)$.
It is the bridge between functors into $\mathbf{Cat}$ and categories fibred over $\mathcal{C}$.
It contains the category of elements of a set-valued functor as a special case.

## Explanations

### L1 human

Think of $\mathcal{C}$ as a base, and above each point $c$ of the base sits a whole category $F(c)$, called the fibre.
The Grothendieck construction puts all the fibres side by side and adds arrows that move from one fibre to another, following arrows of the base.
An object of $\int F$ is a pair: a base object $c$ and an object $x$ of the fibre over $c$.
For a small example, if $F$ is constant with value a category $\mathcal{D}$, then $\int F$ is the [[product category|AFNE9C4NH6]] $\mathcal{C} \times \mathcal{D}$.
If every fibre is a set with only identity arrows, the construction gives the category of elements of $F$.