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# All Questions

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### The model category structure on $\mathbf{TMon}$

I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer. I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...
1answer
135 views

### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
1answer
200 views

### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
0answers
58 views

### On cofibrations of simplicially enriched categories

Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells. We have a canonical inclusion functor , i: C \...
1answer
291 views

### Definition A.3.1.5 of Higher Topos Theory

I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...
0answers
136 views

### Why is the Straightening functor the analogue of the Grothendieck construction?

In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...
1answer
245 views

### Does the Dwyer-Kan model structure make dgCat a model $2$-category?

Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category. Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
2answers
645 views

### Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
1answer
127 views

### Inductive folk model structure on strict ω-categories

There is a paper of Lafont, Metayer, and Worytkiewicz  that constructs a model structure on the category of strict $\omega$-categories that they call the folk model structure. This model structure ...
2answers
298 views

### Are cofibrations accessible?

The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations? More generally, let $C$ be a locally presentable ...
1answer
288 views

### Is there an “injective version” of the Bergner model structure?

The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions. ...
0answers
54 views

### Calculating the intersection of the saturations of a decreasing sequence of morphisms

I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm ...
3answers
573 views

### Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories?

Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences. Let $\mathbf Q$ be the corresponding $\infty$-...
1answer
309 views

### Weak complicial sets: Are the morphisms too strict?

In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...
2answers
473 views

### What is a model category from an $\infty$ point of view?

A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example: Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense ...

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