17

The work on analytic geometry is all joint with Dustin Clausen!
Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at the nCatCafe. Also, as David Corfield comments, much of this had been observed long before: https://nforum.ncatlab.org/discussion/5473/etale-site/?Focus=43431#...

12

Since the dual of an abelian category is also an abelian category, the question is equivalent to the same question for projective resolutions.
I will show that the category $\mathbf{Ab}^{\operatorname{f.t.}}$ of finitely generated abelian groups has enough projectives, but no functorial projective cover. The idea is that multiplication by any $c \in \mathbf ...

8

The usual Grothendieck construction has for $\mathcal C$ an ordinary category, so it doesn't have any 2-cells (or at least, doesn't have any non-identity 2-cells). Moreover, we actually get not only a category out of it, but an object of the slice (2-)category $\mathcal Cat/\mathcal C$. If $\mathcal C$ weren't an ordinary category, this wouldn't make as much ...

4

We have a case of relative cohesion used in an algebraic geometric setting discussed at the nLab. The entry for differential algebraic K-theory interprets
Ulrich Bunke, Georg Tamme, Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)
via cohesion over the base $Sh_\infty\left(Sch_{\mathbb{Z}}\right)$, ∞-...

3

It produces a functor between categories. In fact, what is called a fibred category. The construction is detailed in Volume 2 of Borceux's Handbook on Categorical Algebra.
Also Angelo Vistolis Notes on Grothendieck topologies, fibered categories and descent theory is worth looking at.

3

This perspective seems to be absent so far, even though this is a very old question.
To properly credit the source for this idea, the perspective seems to be taken for granted in MacLane and Moerdijk's Sheaves in Geometry and Logic, which was where I was introduced to it.
The observation is the following: Categories are generalizations of monoids, and ...

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