nLab final lift

Final and initial lifts

Final and initial lifts


Let U:CDU\colon C\to D be a functor.


A UU-structured sink is a sink of the form {f i:U(A i)Y}\{f_i \colon U(A_i) \to Y\} in DD.

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

A lift of YY along a UU-structured sink {f i:U(A i)Y}\{f_i\colon U(A_i)\to Y\} is an object BB of CC, equipped with a sink {ϕ i:A iB}\{\phi_i\colon A_i \to B\} in CC and a morphism h:YU(B)h\colon Y\to U(B) in DD, such that U(ϕ i)=hf iU(\phi_i) = h \circ f_i for each ii. A morphism of lifts is, of course, a morphism ξ:BB\xi\colon B\to B' of CC such that the sink {ϕ i:A iB}\{\phi'_i\colon A_i\to B'\} factors through the sink {ϕ i:A iB}\{\phi_i\colon A_i\to B\} as ϕ i=ξϕ i\phi_i'=\xi\circ\phi_i, and such that the morphism h:YU(B)h'\colon Y\to U(B') factors through the morphism h:YU(B)h\colon Y\to U(B) as h=p(ξ)hh'=p(\xi)\circ h.

Note that if UU is faithful, then it suffices to demand merely that the sink {ϕ i:A iB}\{\phi_i\colon A_i \to B\} exists, rather than giving it as part of the structure.


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

If objects of CC are regarded as objects of DD equipped with structure, for a (strictly) final lift we say that BB is the final structure or strong structure on YY induced by the sink. Note that if UU 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 DD is the terminal category, then a UU-structured sink is simply a family of objects A iA_i of CC, a lift is just a cocone A iBA_i\to B, a morphism of lifts is a morphism of cocones, hence a semi-final lift is simply a colimit in CC.

  • An empty UU-structured sink is just an object YY of DD, a lift of it is a morphism YU(B)Y\to U(B), and a morphism of lifts is a morphism BBB\to B' so that YU(B)Y\to U(B') factors as YU(B)U(B)Y\to U(B)\to U(B'). Hence, a semi-final lift of such a sink is a free object with unit morphism h:YU(B)h\colon Y\to U(B) of DD. Thus UU 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 UU-structured sink is just a morphism of the form f:U(X)Yf\colon U(X) \to Y. A strictly final lift of such a sink is precisely an opcartesian arrow lying over ff. Thus UU 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:TopSetU\colon Top \to Set.

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

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


Last revised on February 22, 2021 at 19:33:40. See the history of this page for a list of all contributions to it.