Limitations on model-categorical presentations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:47:21Z http://mathoverflow.net/feeds/question/95513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95513/limitations-on-model-categorical-presentations Limitations on model-categorical presentations Mike Shulman 2012-04-29T17:18:06Z 2012-05-07T22:52:16Z <p>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.</p> <p>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:</p> <blockquote> <p>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?</p> </blockquote> <p>But I would be interested in answers to any similar question.</p> http://mathoverflow.net/questions/95513/limitations-on-model-categorical-presentations/95660#95660 Answer by Daniel Schäppi for Limitations on model-categorical presentations Daniel Schäppi 2012-05-01T11:02:24Z 2012-05-01T11:02:24Z <p>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: </p> <p>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.</p> <p>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:</p> <p>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 $\mathbf{sSet}$.</p> <p>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). </p> <p>The above two facts taken together contradict Remark 11.2 of the paper</p> <p>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</p> <p>where this is deduced from a result due to Lewis (Theorem 11.1 in the above paper).</p> http://mathoverflow.net/questions/95513/limitations-on-model-categorical-presentations/96276#96276 Answer by Mike Shulman for Limitations on model-categorical presentations Mike Shulman 2012-05-07T22:52:16Z 2012-05-07T22:52:16Z <p>Here is another answer that involves adding extra properties. If we have a model category which </p> <ul> <li>is locally cartesian closed, as a category (such as if it is a presheaf category)</li> <li>has its cofibrations being the monomorphisms (hence in particular all objects are cofibrant)</li> <li>is right proper (such as if all objects are fibrant)</li> </ul> <p>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.</p> <p>Thus, an $(\infty,1)$-category which is <em>not</em> locally cartesian closed cannot be presented by a model category with all three of the above properties.</p>