nLab final lift

Definition

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

Definition

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

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

A lift of $Y$ along a $U$-structured sink $\{f_i\colon U(A_i)\to Y\}$ is an object $B$ of $C$, equipped with a sink $\{\phi_i\colon A_i \to B\}$ in $C$ and a morphism $h\colon Y\to U(B)$ in $D$, such that $U(\phi_i) = h \circ f_i$ for each $i$. A morphism of lifts is, of course, a morphism $\xi\colon B\to B'$ of $C$ such that the sink $\{\phi'_i\colon A_i\to B'\}$ factors through the sink $\{\phi_i\colon A_i\to B\}$ as $\phi_i'=\xi\circ\phi_i$, and such that the morphism $h'\colon Y\to U(B')$ factors through the morphism $h\colon Y\to U(B)$ as $h'=p(\xi)\circ h$.

Note that if $U$ is faithful, then it suffices to demand merely that the sink $\{\phi_i\colon A_i \to B\}$ exists, rather than giving it as part of the structure.

Definition

A semi-final lift of $\{f_i \colon U(X_i) \to Y\}$ is a lift $B$ of $Y$ admitting a unique morphism of lifts to any other lift of $Y$. If the morphism $h\colon Y\to U(B)$ is an isomorphism, then the semi-final lift is called a final lift. If $h$ is an identity, we call $B$ a strictly final lift.

If objects of $C$ are regarded as objects of $D$ equipped with structure, for a (strictly) final lift we say that $B$ 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 a $U$-structured sink is simply a family of objects $A_i$ of $C$, a lift is just a cocone $A_i\to B$, a morphism of lifts is a morphism of cocones, hence a semi-final lift is simply a colimit in $C$.

• An empty $U$-structured sink is just an object $Y$ of $D$, a lift of it is a morphism $Y\to U(B)$, and a morphism of lifts is a morphism $B\to B'$ so that $Y\to U(B')$ factors as $Y\to U(B)\to U(B')$. Hence, a semi-final lift of such a sink is a free object with unit morphism $h\colon Y\to U(B)$ of $D$. 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. if it admits discrete objects).

• A singleton $U$-structured sink is just a morphism of the form $f\colon U(X) \to Y$. 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 $U\colon Top \to Set$.

• 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.