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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 …

Given that Sp is better behaved than all other existing models of spectra No, Sp is not better behaved than other models. The reason that it seems to be is because all operations in Sp (e.g., Ω^∞ …

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 …

Consider the canonical functor $H$ from the homotopy category of homotopy coherent descent data in the ∞-category of coherent sheaves to the category of descent data in the derived category of coheren …

Various generalizations of the Brown representability theorem found in the literature identify additional conditions one can impose on a category $C$ so that functors $\def\op{{\sf op}}\def\Set{{\sf S …

Categories (co)fibered in groupoids are used to define stacks (stacks in groupoids, to be precise), so any introduction to stacks will do. For stacks on smooth manifolds, see, for example, Ieke Moerd …

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 …

Kan simplicial manifolds are in the same relation to differentiable ∞-stacks (i.e., locally fibrant simplicial presheaves on the site of cartesian spaces and smooth maps) as smooth manifolds are to sh …

The most obvious approach is to consider simplicial $\def\Ai{{\sf A}_∞}\Ai$-categories, where $\Ai$ denotes a nonsymmetric operad in simplicial sets that is weakly equivalent to the terminal operad, i …

The case when $M$ is a smooth manifold follows from the smooth Oka principle. See there for an expository account of the argument and references to additional sources. Indeed, the left side of (*) is …

This is closely related to the Baez–Dolan stabilization hypothesis. There are numerous proofs of this statement. One line of reasoning is to establish a general 1-category statement first: given a sym …

More precisely, let A,B be two simplicially enriched model categories, is it true that every adjoint pair of ∞-functors between A,B comes from a Quillen pair? Assuming the model categories are combi …

Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical construct …

I would say that understanding traditional differential cohomology is a reasonable prerequisite. There are multiple good sources: Ulrich Bunke: Differential cohomology Diferential Cohomology. Categ …