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Funny that someone upped this question of mine today. After ten years I can say that yes: there are at least three examples of "model bicategory", each irreducible to one another, but in a sense diffe …

This stream of thought led to the following preprint that has been on the arXiv for a few days. https://arxiv.org/abs/1704.00303 This is not an act of self-promotion, but maybe some of the interested …

I am looking for an answer to the following question: Let $F : {\cal C} \to {\cal D}$ be a left Quillen functor between combinatorial model categories; let $\tilde F \dashv \tilde G$ be the induced a …

I say that a category of (say) algebras for a monad[¹] $\text{Alg}(\mathbb T)$ is "uninteresting" if the only model structures on $\text{Alg}(\mathbb T)$ result as transfer of the nine model structure …

In the paper Cordier, Jean-Marc, and Timothy Porter. "Homotopy coherent category theory." Transactions of the American Mathematical Society 349.1 (1997): 1-54. the authors define a notion of coh …

There is this old result by Freyd that "homotopy is not concrete": Freyd, Peter. "Homotopy is not concrete." The Steenrod Algebra and Its Applications: A Conference to Celebrate NE Steenrod's Six …

one can always define a t-structure on a stable model category as a t-structure on its homotopy category This is a definition, but it's extremely unsatisfying and completely unenlightening. The …

From a conceptual point of view, the notion of a "model bicategory" is clear: a complete, cocomplete bicategory where there are two "very weak" factorization systems, where the commutativity of the sq …

Hovey's book seems exactly what you need. But take a look at http://folk.uio.no/paularne/SUPh05/DS.pdf which is nothing but a self-contained rewriting of Quillen's original work "Homotopical Algebra"; …

(Sorry for the crossposting, but I'm really interested in this question). One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological spac …

In the hope this won't turn into a trivial problem (I couldn't find a similar discussion here), here's my question. I'm studying a little homotopical algebra in this article by Brown. You can easily …