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      Questions tagged [topos-theory]

      The tag has no usage guidance.

      11
      votes
      0answers
      128 views

      Ordinal-valued sheaves as internal ordinals

      Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\...
      5
      votes
      0answers
      192 views

      Roadmap to understand gerbe in the sense of Lurie’s Higher Topos Theory

      Definition $7.2.2.20$ : Let $\mathfrak{X} $ be an $\infty$-topos. An $n$-gerbe on $\mathfrak{X}$ is an object in $\mathfrak{X}$ which is $n$-connective and $n$-truncated. Above is the definition of ...
      5
      votes
      1answer
      183 views

      Smash product and the integers in a Grothendieck $(\infty, 1)$-topos

      Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to this page in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \...
      3
      votes
      0answers
      150 views

      What category of toposes is monadic over the 2-category of groupoids?

      Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the ...
      17
      votes
      1answer
      657 views

      Commutative rings : Topoi = Fields :?

      The following is probably a bad question, but hopefully, it might have a very good answer. In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...
      12
      votes
      1answer
      242 views

      Is Set a finitely presentable object in Topoi?

      The setting for this question is the (2,1) category Topoi. 0-cells in Topoi are Grothendieck topoi. 1-cells are geometric morphisms and have the direction of the right adjoint. 2-cells are invertible ...
      5
      votes
      1answer
      191 views

      Automorphism group of a torsor

      Given a site $C$ and an object $U$, let $G$ be a sheaf of groups on this site and let $F$ be $G$-torsor, see the Stacks Project for the general definition. By restriction on the over category $C/U$ (...
      9
      votes
      0answers
      202 views

      Adjoints to forcing

      Forcing over a partial order $P$ can be viewed in a category theoretic sense as constructing the presheaf topos ${\bf Set}^{P^{op}}$ over the partial order (viewed as a category) then passing through ...
      10
      votes
      0answers
      624 views

      Toposophy vs Set theoretical multiverse philosophy

      Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
      5
      votes
      1answer
      305 views

      Universal property of sheaf category

      Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
      2
      votes
      1answer
      83 views

      Definition of $\in_c$ for power objects

      On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...
      5
      votes
      0answers
      94 views

      Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)

      SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
      5
      votes
      0answers
      121 views

      Topos with enough projectives

      It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
      8
      votes
      1answer
      197 views

      Is there a version of the “infinitary” disjunctive normal form theorem for topoi and slice categories?

      According to nLab Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra. This is basically an infinitary ...
      8
      votes
      2answers
      335 views

      What is a spectrum object in $\infty$-topoi?

      For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra? To ...

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