Let be a functor.
A -structured sink is a sink of the form in .
Note that like all sinks, a -structured sink is not necessarily assumed to be small.
A lift of along a -structured sink is an object of , equipped with a sink in and a morphism in , such that for each . A morphism of lifts is, of course, a morphism of such that the sink factors through the sink as , and such that the morphism factors through the morphism as .
Note that if is faithful, then it suffices to demand merely that the sink exists, rather than giving it as part of the structure.
A semi-final lift of is a lift of admitting a unique morphism of lifts to any other lift of . If the morphism is an isomorphism, then the semi-final lift is called a final lift. If is an identity, we call a strictly final lift.
If objects of are regarded as objects of equipped with structure, for a (strictly) final lift we say that is the final structure or strong structure on induced by the sink. Note that if is an isofibration, then any final lift may be made into a strictly final one.
The dual concept, which applies to cosinks (“sources”), is called a (perhaps semi- or strictly) initial lift, an initial structure or a weak structure.
If is the terminal category, then a -structured sink is simply a family of objects of , a lift is just a cocone , a morphism of lifts is a morphism of cocones, hence a semi-final lift is simply a colimit in .
An empty -structured sink is just an object of , a lift of it is a morphism , and a morphism of lifts is a morphism so that factors as . Hence, a semi-final lift of such a sink is a free object with unit morphism of . Thus admits semi-final lifts of empty sinks precisely when it has a left adjoint. Similarly, it admits final lifts of empty sinks precisely when it has a fully faithful left adjoint (i.e. if it admits discrete objects).
A singleton -structured sink is just a morphism of the form . A strictly final lift of such a sink is precisely an opcartesian arrow lying over . Thus admits strictly final lifts of singleton structured sinks precisely when it is a Grothendieck opfibration (and final lifts of such sinks precisely when it is a Street opfibration).
A topological concrete category is a functor that admits final lifts of all (not necessarily small) structured sinks. This turns out to be equivalent to admitting initial lifts of all structured cosinks. The most famous example is then initial topologies and final topologies for .
More generally, a solid functor is one that admits semi-final lifts of all structured sinks.
If has both a left and right adjoint, of which one (and hence also the other) is fully faithful, and is cocomplete, then admits final lifts of all small structured sinks. See adjoint triple for a proof. Dually, if is complete in this situation, then admits initial lifts of all small structured cosinks.
(Semi-)final lifts can be generalized to (semi-)final extensions, which are to (semi-)final lifts as Kan extensions are to colimits.
In Higher Topos Theory (section 4.3.1) the corresponding notion of (strictly) final lift for (∞,1)-categories is called a -colimit.
Last revised on February 22, 2021 at 19:33:40. See the history of this page for a list of all contributions to it.