A solid functor (also called a semi-topological functor) is a forgetful functor $U\colon A\to X$ for which the structure of an $A$-object can be universally lifted along sinks. One can also say that $U$ has not just a left adjoint but all possible “relative” left adjoints.
When $U$ is solid, colimits in $A$ can be constructed in a natural way out of colimits in $X$, and $A$ inherits strong cocompleteness properties from $X$.
Let $U\colon A\to X$ be a faithful functor. A $U$-structured sink is a sink in $X$ of the form $(U a_i \overset{f_i}{\to} x)$. Note that the indexing family $i\in I$ need not be a set, it can be a proper class. A semi-final lift of such a $U$-structured sink consists of a morphism $x\overset{g}{\to} U b$ in $X$ such that
Every composite $g \circ f_i\colon U a_i \to U b$ is in the image of $U$, i.e. is of the form $U(\tilde{g})$ for some $\tilde{g}\colon a_i\to b$ (necessarily unique, since $U$ is faithful), and
$g$ is initial with this property, i.e. for any other morphism $x \overset{g'}{\to} U b'$ such that each $g' \circ f_i$ is in the image of $U$, there exists a unique $h\colon b\to b'$ in $A$ such that $g' = U(h) \circ g$.
Finally, $U$ is called solid if every $U$-structured sink has a semi-final lift.
Any topological functor is solid. Indeed, a functor $U$ is topological just when it has final lifts for all $U$-structured sinks, where a final lift is a semi-final lift for which $g$ is an isomorphism.
Any monadic functor into $Set$ is solid.
A fully faithful functor is solid if and only if it has a left adjoint.
If $U\colon A\to X$ is faithful and has a left adjoint, and moreover $A$ is cocomplete and well-copowered, then $U$ is solid.
For $C$ a cofibrantly generated model category with monic generating cofibrations, the forgetful functor from algebraic fibrant objects to $C$ is solid. See there for details.
For any $x\in X$, the empty family of morphisms into $x$ is a $U$-structured sink, and a semi-final lift for this family is a universal arrow $x\to U b$. Therefore, if $U$ is solid, then it has a left adjoint.
Suppose that $U\colon A\to X$ is solid and let $F\colon D\to A$ be a diagram such that $U F$ has a colimit in $X$, consisting of a cocone $U F d_i \to c$. Let $c \to U e$ be a semi-final lift of this $U$-structured sink, for which we have induced morphisms $F d_i \to e$ in $A$. Since $U$ is faithful, these morphisms are a cocone under $F$, and the semi-finality makes it into a colimit in $A$.
Therefore, if $A$ is solid over $X$, then it admits all colimits which $X$ does. Moreover, if we understand colimits in $X$, and we understand the semi-final lifts, then we understand colimits in $A$.
In particular, if $X$ is cocomplete, then so is $A$. In fact, more is true: if $X$ is total, then so is $A$.
The standard transfer theorem for model structures states that if $U\colon A\to X$ is a functor such that
then $A$ has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by $U$. Using an argument of (Nikolaus) we can show:
Let $U\colon A\to X$ be an accessible solid functor, and assume that $X$ has a cofibrantly generated model structure and the following acyclicity condition:
Then the transfer theorem applies, so that $A$ has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by $U$.
We have remarked above that $U$ has a left adjoint, and we assumed it to be accessible, so it remains to show that the given acyclicity condition implies the standard one.
We first show that pushouts in $A$ of images under $F$ of generating acyclic cofibrations become acyclic cofibrations (not just weak equivalences) upon applying $U$. Let $i\colon x\to y$ be a generating acyclic cofibration, and
a diagram in $A$ of which we would like to take the pushout. Consider the pushout of the corresponding diagram in $X$:
Since $X$ is a model category, $g$ is an acyclic cofibration. Therefore, if $U(a) \sqcup_x y \overset{k}{\to} U(b)$ is a semifinal lift of the singleton sink $\{g\}$, by assumption, $k$ is also an acyclic cofibration and thus so is the composite $U(a)\to U(b)$. But it is straightforward to verify that in fact, the map $a\to b$ of which this is the image (which exists by assumption) gives a pushout diagram in $A$:
If $U$ is not just accessible but finitary, then it preserves all transfinite composites, so any transfinite composite of such pushouts in $A$ maps to a transfinite composite in $X$, and we know that transfinite composites of acyclic cofibrations in $X$ are acyclic cofibrations, so the desired acyclicity condition follows. In general, we can argue as follows: given a transfinite sequence $a_0\to a_1\to\dots$ in $A$ of such pushouts, its colimit (= composition) can be constructed as above by forgetting down to $X$, taking the colimit there, and then taking the semifinal lift. But since acyclic cofibrations in $X$ are closed under transfinite composites, the legs of the colimiting cocone in $X$ are acyclic cofibrations. Hence by the assumed acyclicity condition, so is the semifinal lift, and hence (by composition) so are the images in $X$ of the legs of the colimiting cone in $A$. This completes the proof.
Walter Tholen, Semitopological functors. I
and others…
The example of algebraic fibrant objects and the argument entering the above lifting theorem appears in
See also model structure on algebraic fibrant objects.