nLab
final lift

Definition

Let $U:C\to D$ be a functor.

Definition

A $U$-structured sink is a sink of the form $\{f_i:U(X_i)\to Y\}$ in $D$.

Note that like all sinks, a $U$-structured sink is not necessarily assumed to be small.

Given such a structured sink, let $\{f_i\}/D$ denote the category whose objects are objects $Z\in C$ equipped with a morphism $h:Y\to U(Z)$ in $D$ and a sink $\{g_i:X_i\to Z\}$ in $C$, such that $U(g_i)=h\circ f_i$ for all $i$. (Its morphisms are, of course, morphisms in $C$ commuting with the structure.)

Note that if $U$ is faithful, then it suffices to demand merely that the $g_i$ exist, rather than giving them as part of the structure.

Definition

A semi-final lift of $\{f_i:U(X_i)\to Y\}$ is an initial object of $\{f_i\}/D$. If $h$ is an isomorphism, then it is called a final lift. If $h$ is an identity, we call it a strictly final lift.

If objects of $C$ are regarded as objects of $D$ equipped with structure, in a (strictly) final lift we say that $Z$ is the final structure or strong structure on $Y$ induced by the sink. Note that if $U$ 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.

Examples

• If $D$ is the terminal category, then final (or semi-final) lifts reduce to simply colimits in $C$.

• An empty $U$-structured sink is just an object of $D$, and a semi-final lift of such a sink is a free object on $D$. Thus $U$ 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. it admits discrete objects).

• A singleton $U$-structured sink is just a morphism of the form $f:U(X)\to Y$. A strictly final lift of such a sink is precisely an opcartesian arrow lying over $f$. Thus $U$ 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 $U:\mathrm{Top}\to \mathrm{Set}$.

• More generally, a solid functor is one that admits semi-final lifts of all structured sinks.

• If $U$ has both a left and right adjoint, of which one (and hence also the other) is fully faithful, and $C$ is cocomplete, then $U$ admits final lifts of all small structured sinks. See adjoint triple for a proof. Dually, if $C$ is complete in this situation, then $U$ admits initial lifts of all small structured cosinks.

Remarks

• (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 $U$-colimit.