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 $F^*(g)$ assigns $b$ 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 non-evil thing on this page 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 May 28, 2021 at 12:17:07. See the history of this page for a list of all contributions to it.