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A topos is a category that behaves very much like the category of sets and possesses a good notion of localization. Related to topos are: sheaves, presheaves, descent, stacks, localization,...

29 votes
2 answers
4k views

The philosophy behind local rings

This question has been bugging me for a while and I can't seem to make sense of it on a clear conceptual level. The theory of local rings is given by taking the theory of rings and adding the axioms …
Georg Lehner's user avatar
  • 2,303
25 votes
1 answer
2k views

A geometric theory of Blueprints? (Algebras over the field with one element)

In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a l …
Georg Lehner's user avatar
  • 2,303
24 votes
1 answer
1k views

Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology

An $(\infty,1)$-topos according to Lurie is defined as (accessible) left exact localization of a presheaf $(\infty,1)$-category $\text{P}(\mathcal C)$. Those $(\infty,1)$-topoi $\text{Sh}(\mathcal C)$ …
Georg Lehner's user avatar
  • 2,303
7 votes
1 answer
161 views

When is the category of sheaves on a site compactly assembled/a continuous category?

If $(C,J)$ is a site, what is a natural condition on the Grothendieck topology $J$ to ensure that the category $Sh(C,J)$ is compactly assembled? I am both interested in the 1-categorical as well as th …
Georg Lehner's user avatar
  • 2,303
5 votes
0 answers
157 views

Resources on a smooth topos containing complex analytic/holomorphic geometry

In this question Urs Schreiber mentioned there are models in synthetic differential geometry of complex analytic geometry. First of all: When Urs writes complex analytic geometry, does he mean comple …
Georg Lehner's user avatar
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