Questions tagged [topos-theory]
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,...
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Direct and inverse images of internal derived Hom sheafs
Let $i_{\ast}, i^{\ast}: Sh(X) \to Sh(Y)$ be a geometric morphism of topos.
In the derived category $D(X)$ of abelian sheaves on $X$, we can consider the internal derived Hom: $R\mathcal{Hom}_{D(X)}(F,...
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Counterexample of cohomological descent for unbounded complexes
In [BS15, Proposition 3.3.6], [LO08, 2.2], and Stacks Project 0DC1, cohomological descent for unbounded complexes is proved under the assumptions of left-completeness or finiteness assumptions on the ...
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What alternative set theories are topoi?
The category of sets in set theory in the incarnation known as ZF forms a topos, it is the archetypal topos and I think the motivation for elaborating the theory of topoi.
Now there are other ...
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Recommendations for Sheaf and Topos Theory with a View Towards Homotopy Theory
I’m an undergraduate student looking to integrate some sheaf and topos theory into my study plan for the summer. my goal is to get into more modern/abstract homotopy theory, but i also know there is a ...
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Is “internal CoSHEP” the same as CoSHEP in stack semantics?
Recall that an object $M$ in a Heyting pretopos is said to be internally projective if $M$ is projective in the stack semantics. There are various other ways of expressing the notion of “internally ...
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Logical differences between Effective and Kleene-Vesley topos
The Kleene-Vesley topos $RT(\mathcal{K}_2,\mathcal{K}_2^{rec})$ and Effective topos $RT(\mathcal{K_1})$ share quite a few properties. They're evidently both constructive and reject LPO, and both have ...
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Is higher topos theory valid for non-Hausdorff spaces
My question is whether, in higher topos theory, Lurie supposes Hausdorff conditions on topological spaces $X$ without saying so or whether I'm missing something and we don't need such conditions:
At ...
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What are some open problems about the topos $G$-$\mathbf{Set}$?
The topos $G$-$\mathbf{Set}$ is the category of left $G$-actions for a fixed group $G$. (The morphisms are what you would expect.)
What are some open problems about $G$-$\mathbf{Set}$?
(Here $G$ need ...
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Equivalence among $\tau$-theory, elementary topos and Mitchell-Bénabou language
In Johnstone's book "Sketches of an Elephant: A topos theory compendium, volume 2" (referred as Elephant), he defined a higher-order typed (intuitionistic) signature (simplified as $\tau$-...
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Why is it that higher-order logic and geometric logic are so tightly connected?
By higher order logic, I mean logic which quantifies freely over types including propositions and functions (thus predicates, predicates of predicates, etc).
We have a direct connection between higher-...
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Conditions on the unit and coünit of a geometric morphism
Let $f = (f^* \dashv f_*) \colon \mathscr{F} \to \mathscr{E}$ be a geometric morphism between topoi. Call $\eta \colon 1_{\mathscr{E}} \to f_* f^*$ the unit and $\varepsilon \colon f^* f_* \to 1_{\...
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Regarding the realizability topos on the computable part of Kleene's second algebra
Let:
$\mathcal{K}_1$ be the first Kleene algebra, meaning $\mathbb{N}$ endowed with the partial operation $(p,n) \mapsto p\bullet n := \varphi_p(n)$ where $\varphi$ is the $p$-th partial computable ...
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Stable analog of topoi
I am wondering about what the appropriate analog of topoi are in the stable setting.
Recall that an $\infty$-topos is a presentable $\infty$-category that is the left-exact reflective localization of ...
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Relating sheaves of categories to categories of sheaves?
Oftentimes when working in the internal logic of a (ringed) topos $X$ (I'm interested in both the $1$-topos and the $\infty$-topos cases), I've found myself wanting to relate the following objects, in ...
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Gaps in the category theory literature you'd like to see filled
Lately, I've been working on the Clowder Project, a crowdfunded category theory wiki and reference work, which aims to become essentially a Stacks Project for category theory.
Part of the reason I ...