nLab
codomain fibration

Definition

For $C$ any category, there is a functor

$$t : [I,C] \to C \,,$$

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 opfibration. It corresponds under the Grothendieck construction to the pseudofunctor

$$C/(-) : C \to Cat$$

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

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

This functor $cod : [I,C] \to C$ 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.

Details

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

The arrow category

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

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

with the obvious composition.

The functor

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 $\iota_1 : {b} \to \{a \to b\}$

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

The op-cartesian lifts

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

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

The cartesian lifts

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

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

Direct image operation

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 $f$.

Inverse image operation

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.

Adjointness of direct and inverse image

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.

Monadic descent

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.

The Subobject Fibration

By restricting our attention to a subset of arrows in the codomain fibration and using the notion of the skeleton of a fibration, we can 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 $cod:\mathcal{C}^\to\to\mathcal{C}$ on a category $\mathcal{C}$ with pullbacks, 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.

In higher category theory

We discuss the codomain fibration in higher category theory.

In 2-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.

In $(\infty,1)$-category theory

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

Proposition

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.

As a universe

(…)

$\mathbf{H}$ 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”.

(…)

Related concepts

• tangent category

• tangent (infinity,1)-category

References

Secton 2.4.7 of

• Jacob Lurie, Higher Topos Theory