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Yes, the model structure on reduced simplicial sets is cofibrantly generated. An explicit proof of this statement is given by Goerss and Jardine in Simplicial Homotopy Theory, the proof of Proposition …

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 …

One reference is the two papers by Nikolaus–Schreiber–Stevenson: Principal ∞-bundles – General theory Principal ∞-bundles – Presentations In particular, these papers explain the equivalence between G- …

Such a notion of a principal $\def\bC{{\bf C}}\bC$-bundle (when $\bC$ is a topological or simplicial category, or a Segal space) is available in Definition 6.1 of the paper Classifying spaces of infin …

The usual constructions of Grothendieck homotopy theory (as presented by Maltsiniotis and Cisinski) can be easily extended to the setting of relative categories. Recall that given a small category $A$ …

The book 2-dimensional categories by Johnson and Yau does seem to satisfy all of the required conditions: it unravels all definitions in full detail, spelling out the details for 2-categories and natu …

“Diffeology” by Patrick Iglesias-Zemmour is probably the closest match. He develops differential forms and de Rham cohomology, fiber bundles, connections, and symplectic geometry in the language of di …

As indicated in the comments, the notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$ is defined (for example) by Milne in Étale Cohomology as the $p$th right derived functor of the inclusi …

An account of derived functors between ∞-categories equipped with weak equivalences and fibrations can be found in Section 7.5 of Cisinski's Higher Categories and Homotopical Algebra. This setting is …

Generators and relations for the nonextended 2-dimensional bordism category already appear in Robbert Dijkgraaf's 1989 PhD dissertation, see Section 3.2.

This journal published by the Herzen University is not yet available in electronic form. A paper version can be found in multiple libraries, including the National Library of Russia. They will scan pa …

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

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 …

Both cases ($F$ is a sheaf of abelian topological groups or abelian Lie groups) can be treated using the same machinery. The Yoneda embedding embeds abelian Lie groups as a fully faithful subcategory …

$T$ can be formalized as a natural transformation $\def\Vect{{\rm Vect}} \def\Vectc{\Vect_\nabla} \Vectc→Ω^n$ of functors $\def\Man{{\sf Man}} \def\op{{\sf op}} \def\Grpd{{\sf Grpd}} \Man^\op → \Grpd$ …