nLab
solid functor

Contents

Idea

A solid functor (also called a semi-topological functor) is a forgetful functor U:AXU\colon A\to X for which the structure of an AA-object can be universally lifted along sinks. One can also say that UU has not just a left adjoint but all possible “relative” left adjoints.

When UU is solid, colimits in AA can be constructed in a natural way out of colimits in XX, and AA inherits strong cocompleteness properties from XX.

Definition

Let U:AXU\colon A\to X be a faithful functor. A UU-structured sink is a sink in XX of the form (Ua if ix)(U a_i \overset{f_i}{\to} x). Note that the indexing family iIi\in I need not be a set, it can be a proper class. A semi-final lift of such a UU-structured sink consists of a morphism xgUbx\overset{g}{\to} U b in XX such that

  1. Every composite gf i:Ua iUbg \circ f_i\colon U a_i \to U b is in the image of UU, i.e. is of the form U(g˜)U(\tilde{g}) for some g˜:a ib\tilde{g}\colon a_i\to b (necessarily unique, since UU is faithful), and

  2. gg is initial with this property, i.e. for any other morphism xgUbx \overset{g'}{\to} U b' such that each gf ig' \circ f_i is in the image of UU, there exists a unique h:bbh\colon b\to b' in AA such that g=U(h)gg' = U(h) \circ g.

Finally, UU is called solid if every UU-structured sink has a semi-final lift.

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Examples

Properties

Left adjoint

For any xXx\in X, the empty family of morphisms into xx is a UU-structured sink, and a semi-final lift for this family is a universal arrow xUbx\to U b. Therefore, if UU is solid, then it has a left adjoint.

Lifting of colimits

Suppose that U:AXU\colon A\to X is solid and let F:DAF\colon D\to A be a diagram such that UFU F has a colimit in XX, consisting of a cocone UFd icU F d_i \to c. Let cUec \to U e be a semi-final lift of this UU-structured sink, for which we have induced morphisms Fd ieF d_i \to e in AA. Since UU is faithful, these morphisms are a cocone under FF, and the semi-finality makes it into a colimit in AA.

Therefore, if AA is solid over XX, then it admits all colimits which XX does. Moreover, if we understand colimits in XX, and we understand the semi-final lifts, then we understand colimits in AA.

In particular, if XX is cocomplete, then so is AA. In fact, more is true: if XX is total, then so is AA.

Lifting of model structures

The standard transfer theorem for model structures states that if U:AXU\colon A\to X is a functor such that

  1. UU has a left adjoint FF,
  2. UU is accessible, i.e. preserves κ\kappa-filtered colimits for sufficiently large κ\kappa,
  3. XX has a cofibrantly generated model structure, and
  4. Transfinite composites of pushouts of images under FF of generating acyclic cofibrations in XX become weak equivalences after applying UU (the acyclicity condition),

then AA has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by UU. Using an argument of (Nikolaus) we can show:

Theorem

Let U:AXU\colon A\to X be an accessible solid functor, and assume that XX has a cofibrantly generated model structure and the following acyclicity condition:

  • If F:DAF\colon D\to A is a filtered diagram and U(F(d i))xU(F(d_i)) \to x is a cocone under UFU\circ F, each of whose legs is an acyclic cofibration in XX, then the semifinal lift xU(b)x\to U(b) of this UU-structured sink is also an acyclic cofibration.

Then the transfer theorem applies, so that AA has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by UU.

Proof

We have remarked above that UU 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 AA of images under FF of generating acyclic cofibrations become acyclic cofibrations (not just weak equivalences) upon applying UU. Let i:xyi\colon x\to y be a generating acyclic cofibration, and

F(x) f a F(i) F(y)\array{F(x) & \overset{f}{\to} & a\\ ^{F (i)}\downarrow \\ F(y)}

a diagram in AA of which we would like to take the pushout. Consider the pushout of the corresponding diagram in XX:

x f¯ U(a) i g y h U(a) xy.\array{x & \overset{\bar{f}}{\to} & U(a)\\ ^{i}\downarrow && \downarrow^g\\ y& \underset{h}{\to} & U(a) \sqcup_{x} y.}

Since XX is a model category, gg is an acyclic cofibration. Therefore, if U(a) xykU(b)U(a) \sqcup_x y \overset{k}{\to} U(b) is a semifinal lift of the singleton sink {g}\{g\}, by assumption, kk is also an acyclic cofibration and thus so is the composite U(a)U(b)U(a)\to U(b). But it is straightforward to verify that in fact, the map aba\to b of which this is the image (which exists by assumption) gives a pushout diagram in AA:

F(x) f a F(i) F(y) h¯ b.\array{F(x) & \overset{f}{\to} & a\\ ^{F(i)}\downarrow && \downarrow\\ F(y) & \underset{\bar{h}}{\to} & b.}

If UU is not just accessible but finitary, then it preserves all transfinite composites, so any transfinite composite of such pushouts in AA maps to a transfinite composite in XX, and we know that transfinite composites of acyclic cofibrations in XX are acyclic cofibrations, so the desired acyclicity condition follows. In general, we can argue as follows: given a transfinite sequence a 0a 1a_0\to a_1\to\dots in AA of such pushouts, its colimit (= composition) can be constructed as above by forgetting down to XX, taking the colimit there, and then taking the semifinal lift. But since acyclic cofibrations in XX are closed under transfinite composites, the legs of the colimiting cocone in XX are acyclic cofibrations. Hence by the assumed acyclicity condition, so is the semifinal lift, and hence (by composition) so are the images in XX of the legs of the colimiting cone in AA. This completes the proof.

References

The example of algebraic fibrant objects and the argument entering the above lifting theorem appears in

See also model structure on algebraic fibrant objects.

Revised on December 12, 2011 21:05:03 by Mike Shulman (71.136.231.40)