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A good answer to both questions is provided by the following variant of the Gelfand duality for commutative von Neumann algebras, which shows that the following categories are equivalent: The categor …

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 …

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 …

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 …

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 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 …

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 …