(in category theory/type theory/computer science)

**of all homotopy types**

**of (-1)-truncated types/h-propositions**

For $C$ any category, there is a functor

$t \;\colon\; [I,C] \longrightarrow C
\,,$

from the arrow category $[I,C] = Arr(C)$ that sends each morphism $(c_1 \stackrel{f}{\to} c_2) \in [I,C]$ to its codomain $c_2$.

This functor is always an Grothendieck opfibration. Under the Grothendieck construction it corresponds to the pseudofunctor

$C_{(-)} \,\colon\, C \longrightarrow Cat$

that sends each object $c \in C$ to the overcategory $C_{/c}$.

If $C$ has all pullbacks, then the functor is in addition a Grothendieck fibration, hence a bifibration. Traditionally it is this fibered aspect is emphasised (and it even motivates the notion of *Cartesian fibrations*). A right adjoint $u_*$ of $u^*$ exists for every morphism $u$ in $C$ iff C is a locally cartesian closed category.

In any case, this functor $cod \colon [I,C] \to C$ is called the **codomain fibration** of $C$.

Some say **basic fibration** or **self-indexing** or the **fundamental fibration** — anything with so many names must be important!

If instead of the codomain the domain is used, one obtains the dual notion: domain opfibration.

We spell out the details of the functor, of its cartesian and opcartesian morphisms and their properties.

Recall from the discussion at arrow category that the objects in $Arr(C)$ are morphisms in $C$ and the morphisms $(f:x_1\to x_2)\to (g: y_1\to y_2)$ in $Arr(C)$ are the commutative squares in $C$ of the form

$\array{
x_1 &\stackrel{v}\longrightarrow& y_1
\\
\big\downarrow\mathrlap{f}
&&
\big\downarrow\mathrlap{g}
\\
x_2 &\stackrel{u}\longrightarrow& y_2
}$

with the obvious composition.

The functor $cod : Arr(C)\to C$ is given on objects by the codomain (= target) map, and on morphisms it gives the lower arrow of the commutative square.

$cod :
\;\;
\left(
\array{
x_1 &\stackrel{v}\to& y_1
\\
\downarrow\mathrlap{f}
&&
\downarrow\mathrlap{g}
\\
x_2 &\stackrel{u}\to& y_2
}
\right)
\;\;
\mapsto
\;\;
(x_2 \stackrel{u}\to y_2)
\,.$

If we write $[I,C]$ for the arrow category, where $I$ is the interval category $I = \{a \to b\}$, then this functor is the hom-functor applied to the inclusion $\iota_1 : {b} \to \{a \to b\}$

$cod = Hom_\text{Cat}(\iota_1, 1_C) : [I,C] \to [{*}, C] = C
\,.$

That the functor $cod : [I,C] \to C$ is an opfibration means that for each object $\hat c_1 \to c_1$ of $[I,C]$, each morphism $c_1 \stackrel{f}{\to} c_2$ in $C$ has a lift to a morphism

$\array{
\hat c_1 &\to& ??
\\
\downarrow && \downarrow
\\
c_1 &\to& c_2
}$

in $[I,C]$ that is a opcartesian morphism.

Such a lift is given by

$\array{
\hat c_1 &\stackrel{Id}{\to}& \hat c_1
\\
\downarrow && \downarrow
\\
c_1 &\to& c_2
}
\,.$

For given any commuting triangle

$\array{
&& c_2
\\
& \nearrow && \searrow
\\
c_1 &&\to&& c_3
}$

in $C$, and any lift

$\array{
\hat c_1 &\to& d
\\
\downarrow && \downarrow
\\
c_1 &\to& c_3
}$

of $c_1 \to c_3$, there is the unique lift

$\array{
\hat c_1 &\to& d
\\
\downarrow && \downarrow
\\
c_2 &\to& c_3
}$

such that

$\left(
\array{
\hat c_1 &\stackrel{Id}{\to}& \hat c_1
&\to& d
\\
\downarrow && \downarrow && \downarrow
\\
c_1 &\to& c_2 &\to& c_3
}
\right)
\;\;\;
=
\;\;\;
\array{
\hat c_1
&\to& d
\\
\downarrow && \downarrow
\\
c_1 &\to& c_3
}
\,.$

If $C$ has pullbacks, then $cod : [I,C] \to C$ is in addition a fibered category in the sense of Grothendieck:

for every object $\hat c_2 \to c_2$ in $[I,C]$, the cartesian lift of a morphism $c_1 \to c_2$ in $C$ is given by the morphism

$\array{
c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
\\
\downarrow && \downarrow
\\
c_1 &\to& c_2
}
\,.$

Because for

$\array{
&& c_3
\\
& \swarrow && \searrow
\\
c_1 &&\to&& c_2
}$

any commuting triangle in $C$, and for

$\array{
d &\to& \hat c_2
\\
\downarrow && \downarrow
\\
c_3 &\to& c_2
}$

any lift of $c_3 \to c_2$ in $[I,C]$, which by the commutativity of the triangle we may write as

$\array{
d &\to& &\to& \hat c_2
\\
\downarrow && && \downarrow
\\
c_3 &\to& c_1 &\to& c_2
}$

there is, precisely by the universal property of the pullback, a unique morphism, $d\to c_1 \times_{c_2} \hat c_2$ in $C$ such that this factors as

$\array{
d &\to& c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
\\
\downarrow && \downarrow && \downarrow
\\
c_3 &\to& c_1 &\to& c_2
}
\,.$

Recall that in an opfibration $p : E\to B$ , the *direct image* $f_!$ of an object $e \in E$ along a morphism $p(e) \to d$ is the codomain $f_!(e)$ of the opcartesian lift $\hat f : e \to f_! e$ of $f$.

By the above discussion this means that in the codomain opfibration $cod : [I,C] \to C$ the direct image of an object $\hat c_1 \to c_1$ in $[I,C]$ along some morphism $f : c_1 \to c_2$ is the composite morphism $\hat c_1 \to c_1 \to c_2$ in $C$, regarded as an object in $[I,C]$: this yields the functor

$f_! : C/{c_1} \to C/{c_2}$

of overcategories obained by postcomposition with $f$.

Recall that in an fibration $p : E\to B$ , the *inverse image* $f^*$ of an object $e \in E$ along a morphism $d \to p(e)$ is the domain $f^*(e)$ of the cartesian lift $\hat f : f^* e \to e$ of $f$.

By the above discussion this means that in the codomain fibration $cod : [I,C] \to C$ the inverse image of an object $\hat c_2 \to c_2$ in $[I,C]$ along some morphism $f : c_1 \to c_2$ is the morphism out of the pullback $f^* c_2 = c_1 \times_{c_2} \hat c_2 \to c_1$ in $C$, regarded as an object in $[I,C]$: this yields the functor

$C/{c_1} \leftarrow C/{c_2} : f^*$

of overcategories obained by pullback.

For every morphism $f : c_1 \to c_2$ in $C$, the direct and inverse image functors are a pair of adjoint functors

$f_! : C/{c_1} \to C/{c_2} : f^*$

with $f_!$ left adjoint and $f^*$ right adjoint, $f_! \dashv f^*$.

By the above discussion, the adjunction isomorphism

$Hom_{C_2}(f_! \hat c_1, \hat c_2)
\simeq
Hom_{C_1}(\hat c_1, f^*\hat c_2)$

is given by the universal property of the pullback operation, which says that morphisms

$(f_! \hat c_1 \to \hat c_2)
=
\left(
\array{
\hat c_1 &\to& \hat c_2
\\
\downarrow && \downarrow
\\
c_1 &\to& c_2
}
\right)$

factor uniquely through the pullback

$\left(
\array{
\hat c_1 &\to& c_1 \times_{c_2} \hat c_2 &\to& \hat c_2
\\
&\searrow & \downarrow && \downarrow
\\
&& c_1 &\to& c_2
}
\right)$

and hence uniquely correspond to morphisms

$(\hat c_1 \to f^* \hat c_2)
=
\left(
\array{
\hat c_1 &\to& c_1 \times_{c_2} \hat c_2
\\
\downarrow && \downarrow
\\
c_1 &\to& c_2
}
\right)
\,.$

If $C$ is a model category, and $u:c\to d$ a morphism in $C$, we can consider the induced model structure on the overcategories $C/c$, and $C/d$. The adjoint pair

$u_! : C/c \leftrightarrows C/d : u^*$

is then a Quillen pair.

Since the codomain fibration $cod : [I,C] \to C$ is a bifibration when $C$ has all pullbacks, there is a notion of monadic descent in this case. Details on this are at monadic descent for codomain fibrations.

By restricting our attention to a subset of morphisms in the codomain fibration and using the notion of the skeleton of a fibration, we may define a fibration on a category $\mathcal{C}$ with pullbacks called the *subobject fibration* whose fibers are categories of subobjects for objects of $\mathcal{C}$.

Beginning with the codomain fibration $cod \colon \mathcal{C}^\to \longrightarrow \mathcal{C}$ on a category $\mathcal{C}$ with pullbacks (now writing $C^\to$ for the arrow category), we restrict our attention to the subcategory

$Mono(\mathcal{C})\subseteq\mathcal{C}^\to,$

the full subcategory of $\mathcal{C}^\to$ whose objects are monomorphisms in $\mathcal{C}$, called the **monomorphism category of $\mathcal{C}$**. The resulting functor

$cod:Mono(\mathcal{C})\to\mathcal{C}$

is again a fibration since monomorphisms are stable under pullback; we will call this the **monomorphism fibration of $\mathcal{C}$**. The fibers $Mono(\mathcal{C})_X$ for $X\in\mathcal{C}$ are thin categories since parallel monos in a slice category are equal, but they aren’t subobject categories since antisymmetry is only weakly satisfied – objects with antiparallel arrows between them are necessarily isomorphic, but not necessarily equal. To remedy this, we take the fibered skeleton of the monomorphism fibration; briefly, we convert it into an indexed category using the Grothendieck construction, take the skeleton of each index category, then turn it back into a fibration using the Grothendieck construction in the other direction. The resulting fibration is denoted

$cod:Sub(\mathcal{C})\to\mathcal{C}$

and called the **subobject fibration of $\mathcal{C}$**, and the fibers $Sub(\mathcal{C})_X$ are skeletal thin categories, also known as poset categories.

If we take $\mathcal{C}=Set$ then the fibers $Mono(Set)_X$ of the monomorphism fibration are proper classes consisting of all sets isomorphic to subsets of $X$, which isn’t what we want. The fibers $Sub(Set)_X$ consist of one representative from each isomorphism class of sets isomorphic to subsets of $X$, and is thusly isomorphic to the powerset of $X$ viewed as a poset. That is, $Sub(Set)_X\cong\mathcal{P}(X)$ as posets, with equality holding if we choose the right representatives.

We discuss the codomain fibration in higher category theory.

A categorification in dimension 2 (see 2-category theory) is a codomain 2-fibration, whose main example is $Cat^2$ over $Cat$.

Mike Shulman: I still don’t believe that that is a 2-fibration. How do you lift the 2-cells?

David Roberts: How does one lift the 2-cells in a 2-fibration anyway? The case of $Cat^\mathbf{2}\to Cat$ (using weak 2-functors in $Cat^\mathbf{2}$) should in my opinion be an guiding example for this. Although, perhaps it would be better to consider (at least at first) the underlying (2,1)-category or even the (2,1)-category $Gpd$.

Mike Shulman: I think the guiding example of a 2-fibration should actually be $Fib \to Cat$, as in Hermida’s paper. There, you can lift the 2-cells, because in each fibration you can lift the 1-cells.

Let $\mathcal{X}$ be an (∞,1)-category.

The codomain fibration

$Cod : \mathcal{X}^I \to \mathcal{X}$

is an coCartesian fibration.

It is classified under the (∞,1)-Grothendieck construction by

$A \mapsto \mathcal{X}_{/A}
\,,$

where on the right we have the over-(∞,1)-category.

This is a special case of (Lurie, corollary 2.4.7.12).

For $\mathcal{X}$ an (∞,1)-topos, this is the canonical (infinity,2)-sheaf.

(…)

$\mathbf{H}$ an (∞,1)-topos the codomain fibration is the dependent sum

$\sum_{Type} : \mathbf{H}_{/Type} \to \mathbf{H}_{/*} \simeq \mathbf{H}$

where $Type \in \mathbf{H}$ is the object classifier, of some size. This is the internal universe. Since the slice (∞,1)-topos $\mathbf{H}_{/X}$ is the context given by $X$, in a precise sense $\mathbf{H}_{/Type}$ is the “context of the universe”. And so this says that the codomain fibration is the “context of the universe” regarded over the base $\infty$-topos which is the “outermost universe”.

(…)

- Jacob Lurie, Section 2.4.7 of:
*Higher Topos Theory*

Last revised on April 1, 2023 at 16:56:04. See the history of this page for a list of all contributions to it.