# Contents

## Idea

The notion of domain opfibration is dual to that of codomain fibration. See there for more details.

## Definition

Let $C$ be a category and $\mathrm{Arr}\left(C\right)={C}^{2}$ the corresponding arrow category: the objects in $\mathrm{Arr}\left(C\right)$ are morphisms in $C$ and the morphisms $\left(f:x\to x\prime \right)\to \left(g:y\to y\prime \right)$ in $\mathrm{Arr}\left(C\right)$ are the commutative squares of the form

$\begin{array}{ccc}x& \stackrel{v}{\to }& y\\ ↓f& & ↓g\\ x\prime & \stackrel{u}{\to }& y\prime \end{array}$\array{ x &\stackrel{v}\to& y\\ \downarrow\mathrlap{f} &&\downarrow\mathrlap{g}\\ x' &\stackrel{u}\to& y' }

with the obvious composition.

There is a functor $\mathrm{dom}:\mathrm{Arr}\left(C\right)\to C$ given on objects by the domain (= source) map, and on morphisms it gives the upper arrow of the commutative square. If $C$ has pushouts, then this functor is in fact an opfibered (cofibered) category in the sense of Grothendieck, whose pushforward functor ${u}_{*}$ amounts to the usual pushout of $f$ along $u$ in $C$. The fiber over an object $c$ in $C$ is the undercategory $c↓C$. This opfibered category is called the domain opfibration over $C$ (some say the basic opfibration). This notion is dual to the notion of codomain fibration.

## Remarks on notation

Although the pushforward functor in an opfibration is usually written ${u}_{!}$, in the case of the domain opfibration we usually write it as ${u}_{*}$ instead, following the notation of algebraic geometry. Each such functor also has a right adjoint, given by precomposition (just as in the codomain fibration the pullback functors have left adjoints given by postcomposition). Thus, the the domain opfibration is in fact a bifibration, though traditionally its opfibered aspect is emphasised (and it even motivates the notion of cocartesianess for categories over categories). And while the right adjoints in a bifibration are usually written as ${u}^{*}$, for the domain opfibration we write them as ${u}^{!}$, again to conform to usage in algebraic geometry, where the standard string of adjoints is ${u}_{!}⊣{u}^{*}⊣{u}_{*}⊣{u}^{!}$.

Revised on March 31, 2012 17:59:14 by Stephan Alexander Spahn (79.219.112.170)