nLab
domain opfibration

Contents

Idea

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

Definition

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

x v y f g x u y\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 dom:Arr(C)Cdom: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 CC has pushouts, then this functor is in fact an opfibered (cofibered) category in the sense of Grothendieck, whose pushforward functor u *u_* amounts to the usual pushout of ff along uu in CC. The fiber over an object cc in CC is the undercategory cCc\downarrow C. This opfibered category is called the domain opfibration over CC (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 !u_!, in the case of the domain opfibration we usually write it as u *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 *u^*, for the domain opfibration we write them as u !u^!, again to conform to usage in algebraic geometry, where the standard string of adjoints is u !u *u *u !u_! \dashv u^* \dashv u_* \dashv u^!.

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