# Questions tagged [topos-theory]

The topos-theory tag has no usage guidance.

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### 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**

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### 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**

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**1**answer

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### 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**

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### 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**

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**1**answer

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**

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**1**answer

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**

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**1**answer

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### 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**

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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**

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### 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**

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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**

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### 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**

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### 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**

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### 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**

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**1**answer

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**

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**2**answers

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 ...