nLab discrete fibration

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, namely one where each fiber is a set.

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.

Under this equivalence, the representable presheaf on an object $X$ corresponds to the canonical functor $B/X \to B$ from the slice category over $X$.

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.

Model structure for discrete fibrations

Theorem

(Moser–Sarazola, Theorem 2.18.) Suppose $C$ is a category. The slice category $Cat/C$ admits a combinatorial model structure with the following properties.

• Cofibrations are functors that are injective on objects.
• Trivial fibrations are equivalences that are isofibrations.
• Fibrant objects are discrete fibrations $P\to C$.
• Weak equivalences are given by morphisms whose fibrant replacement is a trivial fibration, i.e., an isomorphism.
• Fibrant replacement is induced by the weak factorization system cofibrantly generated by the morphisms $1\colon[0]\to[1]$, $[1]\sqcup_{[0]}[1]\to[1]$ mapping to $C$ in an arbitrary way.

Theorem

(Moser–Sarazola, Theorem 3.9.) Suppose $C$ is a category. There is a Quillen equivalence

$$Cat/C \rightleftarrows Fun(C^{op},Set),$$

where $Cat/C$ is equipped with a model structure for discrete fibrations, $Fun(C^{op},Set)$ is equipped with the projective model structure (weak equivalences are isomorphisms; cofibrations and fibrations are all maps), the right adjoint $Fun(C^{op},Set)\to Cat/C$ is given by the category of elements construction, and the left adjoint adds formal strict base changes to a fibration in order to a get a strict presheaf of sets.

Further properties

• Just as general fibrations, discrete fibrations are stable under pullbacks in Cat.

• Discrete (op)fibrations are the right class of the comprehensive factorisation system.

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 $B$.

Related concepts

• fibration fibered in groupoids

• Grothendieck fibration, Grothendieck construction

References

Review:

• Angelo Vistoli, §3.4 in: Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry – Grothendieck's FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1-104 [ISBN:978-0-8218-4245-4, math.AG/0412512]

Model structure:

• Lyne Moser, Maru Sarazola, A model structure for Grothendieck fibrations, arXiv:2306.11076.

See also:

• Emily Riehl and Fosco Loregian, Categorical notions of fibration, Expositiones Mathematicae 38 4 (2020) 496-514 [arXiv:1806.06129, doi:10.1016/j.exmath.2019.02.004]

A joint generalisation of the notions of discrete fibrations and discrete opfibrations are studied in the following paper under the name unique factorization liftings, i.e. discrete Conduché functors:

• Marta Bunge, Susan Niefield, Exponentiability and single universes, JPAA 148 (2000) 217–250