nLab
codomain fibration

Definition

For $C$ any category, there is a functor

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

from the arrow category $[I,C]=\mathrm{Arr}(C)$ 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

$$C/(-):C\to \mathrm{Cat}$$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 $\mathrm{cod}:[I,C]\to C$ 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 $\mathrm{Arr}(C)$ are morphisms in $C$ and the morphisms $(f:x_1\to x_2)\to (g:y_1\to y_2)$ in $\mathrm{Arr}(C)$ are the commutative squares in $C$ of the form

$$\begin{array}{ccc}{x}_1& \stackrel{v}{\to }& y_1\\ \downarrow f& & \downarrow g\\ x_2& \stackrel{u}{\to }& y_2\end{array}$$\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 $\mathrm{cod}:\mathrm{Arr}(C)\to C$ is given on objects by the codomain (= target) map, and on morphisms it gives the lower arrow of the commutative square.

$$\mathrm{cod}:\left(\begin{array}{ccc}{x}_1& \stackrel{v}{\to }& y_1\\ \downarrow f& & \downarrow g\\ x_2& \stackrel{u}{\to }& y_2\end{array}\right)\mapsto (x_2\stackrel{u}{\to }{y}_2).$$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 ${\iota }_1:b\to \{a\to b\}$

$$\mathrm{cod}={\mathrm{Hom}}_C({\iota }_1,-):[I,C]\to [*,C]=C.$$cod = Hom_C(\iota_1, -) : [I,C] \to [{*}, C] = C \,.

The op-cartesian lifts

That the functor $\mathrm{cod}:[I,C]\to C$ 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

$$\begin{array}{ccc}{\hat{c}}_1& \to & ??\\ \downarrow & & \downarrow \\ c_1& \to & c_2\end{array}$$\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

$$\begin{array}{ccc}{\hat{c}}_1& \stackrel{\mathrm{Id}}{\to }& \hat{c}\\ \downarrow & & \downarrow \\ c_1& \to & c_2\end{array}.$$\array{ \hat c_1 &\stackrel{Id}{\to}& \hat c \\ \downarrow && \downarrow \\ c_1 &\to& c_2 } \,.

For given any commuting triangle

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

in $C$, and any lift

$$\begin{array}{ccc}{\hat{c}}_1& \to & d\\ \downarrow & & \downarrow \\ c_1& \to & c_3\end{array}$$\array{ \hat c_1 &\to& d \\ \downarrow && \downarrow \\ c_1 &\to& c_3 }

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

$$\begin{array}{ccc}{\hat{c}}_1& \to & d\\ \downarrow & & \downarrow \\ c_2& \to & c_3\end{array}$$\array{ \hat c_1 &\to& d \\ \downarrow && \downarrow \\ c_2 &\to& c_3 }

such that

$$\left(\begin{array}{ccccc}{\hat{c}}_1& \stackrel{\mathrm{Id}}{\to }& {\hat{c}}_1& \to & d\\ \downarrow & & \downarrow & & \downarrow \\ c_1& \to & c_2& \to & c_3\end{array}\right)=\begin{array}{ccc}{\hat{c}}_1& \to & d\\ \downarrow & & \downarrow \\ c_1& \to & c_3\end{array}.$$\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 $\mathrm{cod}:[I,C]\to C$ 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

$$\begin{array}{ccc}{c}_1{\times }_{c_2}{\hat{c}}_2& \to & {\hat{c}}_2\\ \downarrow & & \downarrow \\ c_1& \to & c_2\end{array}.$$\array{ c_1 \times_{c_2} \hat c_2 &\to& \hat c_2 \\ \downarrow && \downarrow \\ c_1 &\to& c_2 } \,.

Because for

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

any commuting trigangle in $C$, and for

$$\begin{array}{ccc}d& \to & {\hat{c}}_2\\ \downarrow & & \downarrow \\ c_3& \to & c_2\end{array}$$\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

$$\begin{array}{ccccc}d& \to & & \to & {\hat{c}}_2\\ \downarrow & & & & \downarrow \\ c_3& \to & c_1& \to & c_2\end{array}$$\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

$$\begin{array}{ccccc}d& \to & c_1{\times }_{c_2}{\hat{c}}_2& \to & {\hat{c}}_2\\ \downarrow & & \downarrow & & \downarrow \\ c_3& \to & c_1& \to & c_2\end{array}.$$\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 $\mathrm{cod}:[I,C]\to C$ 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

$$f_{!}:C/c_1\to C/c_2$$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 $\mathrm{cod}:[I,C]\to C$ 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

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

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

with $f_{!}$ left adjoint and $f^{*}$ right adjoint, $f_{!}⊣f^{*}$.

By the above discussion, the adjunction isomorphism

$${\mathrm{Hom}}_{C_2}(f_{!}{\hat{c}}_1,{\hat{c}}_2)\simeq {\mathrm{Hom}}_{C_1}({\hat{c}}_1,f^{*}{\hat{c}}_2)$$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

$$(f_{!}{\hat{c}}_1\to {\hat{c}}_2)=\left(\begin{array}{ccc}{\hat{c}}_1& \to & {\hat{c}}_2\\ \downarrow & & \downarrow \\ c_1& \to & c_2\end{array}\right)$$(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

$$\left(\begin{array}{ccccc}{\hat{c}}_1& \to & c_1{\times }_{c_2}{\hat{c}}_2& \to & {\hat{c}}_2\\ & \searrow & \downarrow & & \downarrow \\ & & c_1& \to & c_2\end{array}\right)$$\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

$$({\hat{c}}_1\to f^{*}{\hat{c}}_2)=\left(\begin{array}{ccc}{\hat{c}}_1& \to & c_1{\times }_{c_2}{\hat{c}}_2\\ \downarrow & & \downarrow \\ c_1& \to & c_2\end{array}\right).$$(\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

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

is then a Quillen pair.

Monadic descent

Since the codomain fibration $\mathrm{cod}:[I,C]\to C$ 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.

Categorification

A categorification in dimension 2 is a codomain 2-fibration, whose main example is ${\mathrm{Cat}}^2$ over $\mathrm{Cat}$.

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 ${\mathrm{Cat}}^2\to \mathrm{Cat}$ (using weak 2-functors in ${\mathrm{Cat}}^2$) 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 $\mathrm{Gpd}$.

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