Search Results

Here is an example that surprised me at some time in the past. Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a $4$-tuple $(V, …

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an inpu …

Here is what André Joyal wrote in an email to me: No, I have not discovered the model structure for quasi-categories in the 1980's. I became interested in quasi-categories (without the name) around 1 …

Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint fun …

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two mode …

Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006, as well as Joyal's own account in The Theory of Quasi-Cate …

Yes, this is always true. Replacing $X$ by its cofibrant replacement if necessary, we can assume $X$ to be cofibrant. In this case, $\def\Hom{\mathop{\rm Hom}} \Hom(X,-)\colon C→V$ is a right Quillen …

Γ-spaces were introduced by Segal in 1969 as models for what can be now described as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective sp …

if we don't assume properness, I don't even see why the first is homotopy-invariant! The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration is always …

There are many ways to define weak equivalences of simplicial sets without referring to topological spaces. A morphism f is a weak equivalence of simplicial sets if and only if one of the following e …

It seems to be a well-known fact that homotopy (co)limits of (co)simplicial diagrams of nonnegatively graded (co)chain complexes in (Grothendieck) abelian categories can be computed by using the Dold- …

This depends on whether one insists on derived functors landing in the original model category (as is the case with modern approaches of Hinich and Dwyer–Hirschhorn–Kan–Smith), or in its homotopy cate …

On the other hand, the category of commutative monoids seems to be much more subtle. The conditions for the existence of a model structure on commutative monoids were worked out by Jacob Lurie aro …

Consider a category C with weak equivalences, e.g., a model category. For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) alon …

The ∞-categorical limits (respectively colimits) are given by the right (respectively left) adjoint of the constant diagram functor $$C→C^I,$$ where $I$ is the indexing category and $C$ is the ∞-categ …