Contents

category theory

# Contents

## Definition

A functor $F: C \to B$ is a discrete fibration if for every object $c$ in $C$, and every morphism of the form $g : b\to F(c)$ in $B$ there is a unique morphism $h : d\to c$ in $C$ such that $F(h) = g$.

A functor $F: C \to B$ is a discrete opfibration if $F^{op}:C^{op}\to B^{op}$ is a discrete fibration.

A discrete fibration is a special case of a Grothendieck fibration.

Given a cartesian category $E$, internal categories $C,B$ in $E$, an internal functor $F: C \to B$ is a discrete fibration of internal categories if the square

$\begin{matrix} C_1 &\stackrel{F_1}\to& B_1\\ d_0\downarrow && \downarrow d_0\\ C_0 &\stackrel{F_0}\to& B_0 \end{matrix}$

is cartesian.

## Discussion via category of elements

Given a discrete fibration $F: C \to B$, define a functor $F^*: B^{op} \to Set$ as follows:

• For $x$ an object of $B$, let $F^*(x)$ be the set of objects $y$ of $C$ such that $F(y) = x$.
• For $g: x \to y$ a morphism of $B$, let $F^*(g): F^*(y) \to F^*(x)$ be the function that maps each element of $F^*(y)$ to the unique $d$ determined by the definition of discrete fibration above.

There is a size issue here, is $F^*(x)$ in fact small? We say that the fibration has small fibres if so; else we must pass to a larger universe when we define Set.

Conversely, give a functor $F^*: B^{op} \to Set$, define a category $C$ and a discrete fibration $F: C \to B$ as follows:

• Let $C$ be the category of elements of the functor $F$; that is:
• an object of $C$ is a pair consisting of an object $x$ of $B$ and an element of $F^*(x)$,
• a morphism from $(x,a)$ to $(y,b)$ in $C$ is a morphism $g: x \to y$ in $B$ such that $F^*(g)$ maps $b$ to $a$.
• The functor from $C$ to $B$ is the obvious forgetful functor.

If you start from $F^*$, construct $C$ and $F$, and then construct a new $F^*$, it will be equal to the original $F^*$. Conversely, if you start with $C$ and $F$, construct $F^*$, and then construct a new $C'$ and $F'$, then there will be an isomorphism of categories between $C$ and $C'$, relative to which $F$ and $F'$ are equal.

## Invariance under equivalence

Note that the definition of fibration refers to equality of morphisms without previously assuming that the sources match, while the construction of $F^*$ from $F$ refers to equality of objects. This is also why we get equality of functors and isomorphism of categories in the immediately preceding paragraph. So the only thing on this page which respects the principle of equivalence is the idea of a functor to Set. That is the fundamental invariant notion; a discrete fibration is just a convenient way of talking about it.

(…)

## Generalization for spans internal to a category

Let $E$ be a cartesian category. A span of internal categories $A\stackrel{p}\leftarrow C\stackrel{q}\to B$ in $Cat(E)$ is called a discrete fibration from $A$ to $B$ if in the diagram

$\begin{matrix} A_0 & \leftarrow & C_l && \\ \downarrow &&\downarrow i_l &&\\ A &\stackrel{p}\leftarrow & C &\stackrel{i_r}\leftarrow & C_r\\ &&q \downarrow && \downarrow \\ && B &\leftarrow & B_0 \end{matrix}$

in which the two squares are the cartesian satisfies the following 3 properties:

• $p\circ i_r : C_1\to A$ is a discrete fibration

• $q\circ i_l: C_l\to B$ is a discrete opfibration

• Let $X$ be defined as the pullback

$\begin{matrix} X & \to & (C_r)_1 \\ \downarrow &&\downarrow \\ (C_l)_1 &\to & C_0 \end{matrix}$

and $j:X\hookrightarrow C_1\times_{C_0} C_1$ the canonical inclusion. Then the morphism $c\circ j : X\to C_1$, where $c: C_1\times_{C_0} C_1\to C_1$ is the composition morphism of internal category $C$, is invertible.

Example. Given internal functors $a : A\to D$ and $b : B\to D$ in $E$, the obvious span $A\leftarrow a\downarrow b\rightarrow B$ is a discrete fibration from $A$ to $B$.

## References

Emily Riehl and Fosco Loregian, Categorical notions of fibration. arXiv:1806.06129

Last revised on December 4, 2021 at 06:07:18. See the history of this page for a list of all contributions to it.