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.
Last revised on December 12, 2011 at 21:05:03. See the history of this page for a list of all contributions to it.