nLab domain opfibration

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 $Arr(C)= C^2$ the corresponding arrow category: the objects in $Arr(C)$ are morphisms in $C$ and the morphisms $(f:x\to x')\to (g:y\to y')$ in $Arr(C)$ are the commutative squares of the form

$$\array{ x &\stackrel{u}\to& y\\ \downarrow\mathrlap{f} &&\downarrow\mathrlap{g}\\ x' &\stackrel{v}\to& y' }$$

with the obvious composition.

There is a functor $dom:Arr(C)\to C$ 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 $c\downarrow C$. 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 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 $u_! \dashv u^* \dashv u_* \dashv u^!$.