Limitations on model-categorical presentations - MathOverflow

Limitations on model-categorical presentations

2012-04-29T17:18:06Z

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is both strict and skeletal, and a tricategory is not (in general) equivalent to one whose units and interchange law are both strict.

Now a model category can be regarded as a particular sort of strictification of an $(\infty,1)$-category. From this perspective, all sorts of questions along the above lines suggest themselves. For concreteness, I'll ask a particular one:

Does there exist a locally presentable $(\infty,1)$-category which (provably) cannot be presented by a model category in which all objects are both fibrant and cofibrant?

But I would be interested in answers to any similar question.

Answer by Daniel Schäppi for Limitations on model-categorical presentations

2012-05-01T11:02:24Z

Unfortunately I don't have an answer to the actual question. If you ask for more than just a model category I think there are examples:

There is no combinatorial symmetric monoidal simplicial model category $\mathcal{S}$ such that $\mathbf{Ho}(\mathcal{S})$ is the stable homotopy category and every object of $\mathcal{S}$ is both fibrant and cofibrant.

Otherwise the forgetful functor $\mathcal{S} \rightarrow \mathbf{sSet}$ given by homming out of the unit object gives a symmetric monoidal Quillen adjunction which seems to have "too many good properties." Here is a sketch:

As a right Quillen adjoint it commutes with the formation of loop spaces (up to equivalence), and it preserves all weak equivalences, hence it should factor through the zero-space functor from $\Omega$-prespectra to $\mathbf{sSet}$.

Since everything is fibrant, it should be possible to transfer the model structure on $\mathcal{S}$ to commutative algebras in $\mathcal{S}$ (use combinatorial for this).

The above two facts taken together contradict Remark 11.2 of the paper

May, J. P. What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra? New topological contexts for Galois theory and algebraic geometry (BIRS 2008), 215–282

where this is deduced from a result due to Lewis (Theorem 11.1 in the above paper).

Answer by Mike Shulman for Limitations on model-categorical presentations

2012-05-07T22:52:16Z

Here is another answer that involves adding extra properties. If we have a model category which

is locally cartesian closed, as a category (such as if it is a presheaf category)
has its cofibrations being the monomorphisms (hence in particular all objects are cofibrant)
is right proper (such as if all objects are fibrant)

then pullback along a fibration $g\colon A\to B$ preserves both cofibrations and acyclic cofibrations, and so the adjunction $g^* \dashv \Pi_g$ is Quillen. Therefore, the $(\infty,1)$-category presented by this model category is locally cartesian closed.

Thus, an $(\infty,1)$-category which is not locally cartesian closed cannot be presented by a model category with all three of the above properties.