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For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.

9 votes

References for the "nerve of an algebraic variety"

One way to understand your question is in the framework of $\mathbf{A}^1$-homotopy theory. This is because your nerve functor is better understood when defined on a cocomplete category like the categ …
AAK's user avatar
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3 votes
Accepted

Descent properties of spaces

For the first problem, as I wrote in the comments, the author is implicitly using the language of $(\infty,1)$-categories, where it does make sense to speak of such a functor. To understand why, you c …
AAK's user avatar
  • 5,901
6 votes
Accepted

Monoidal structure on simplicial sheaves

Yes. This is well-known and can be deduced, for example, from the general statements in the last section of this paper of Barwick. Take $V$ to be the symmetric monoidal model category of simplicial …
AAK's user avatar
  • 5,901