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Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or 2-groupoid? The delooping of the Lie group $\def\B{{\sf B}} \def\U{{\sf U}} \U(1)$ is the Lie 1-groupoid $\B\U(1)$. The shape $\d …

One can do even better than a Quillen adjunction: Theorem 3.8 in A model structure for Grothendieck fibrations establishes a Quillen equivalence between the projective model structure on presheaves of …

The original reference for such results is Proposition 3.3.3 on page 120 in Fabien Morel, Vladimir Voevodsky, A^1-homotopy theory of schemes, Publications mathématiques de l’I.H.É.S., tome 90 (1999), …

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product. Every finite-dimensional ve …

If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group. Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$. Since $\B Ω G≃G$, this ∞-category is eq …

However, every definition I've seen for higher categories assumes that the source of a composite f;g is equal to the source of f (in diagrammatic notation) and the target of f;g is equal to that of g …

Taking loops (or, in categorical language, endomorphisms of the monoidal unit) is commonly seen as a type of decategorification. For example, decategorifying von Neumann algebras produces Hilbert spac …

Yes. As shown in the paper The enriched Thomason model structure on 2-categories, the category of (strict) 2-categories can be equipped with a model structure that makes it Quillen equivalent to the …

This is an answer the question posed in the last paragraph. There is a canonical forgetful functor from the site of open subsets of ${\bf C}^n$ and holomorphic maps to the site of open subsets of ${\b …

The statement is false in its current form: there are left Quillen functors between stable model categories that do not preserve finite limits. However, since ∞-categories are mentioned, presumably wh …

A connection between Dwyer–Kan hammock localizations and adjoints in 2-categories certainly exists. As already mentioned in the comments, as early as 2002, Dawson–Paré–Pronk in “Adjoining adjoints” m …

The answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the singu …

The analogous adjunction for the enriched category of functors can be deduced from the adjunction for the ordinary category of functors using the Yoneda lemma. Indeed, to establish a natural isomorphi …

An important theoretical application is Kan's fibrant replacement functor $\def\Ex{{\sf Ex}}\def\Exi{\Ex^{\sf\infty}}\Exi$, defined as the filtered colimit of functors $\Ex^n$ ($n≥0$), where $\Ex$ is …

I have not yet found the statement of your answer by localization and construction of the sheaf, in your own publications and available texts, or elsewhere. To the best of my knowledge, there is not …