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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,...

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While constructive logic is compatible with classical logic and is sufficient to develop almost all important theorems from classical complex analysis, constructive is also compatible with axioms that ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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This is a naive question, acknowledging a speculative analogy between two formally distinct domains. I aim to explore whether this perspective could be heuristically fruitful. Synthetic Differential ...
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In Lurie's Spectral algebraic geometry I read the the topos $\mathcal{S}\mathrm{hv}_{\mathrm{Nis}}(X)$ if noetherian, finite krull dimension and quasi compact spectral algebraic space is postnikov ...
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I am currently studying sheaves and topos for logic. If $(\mathcal{O}, \subset)$ is seen as a category. Then if I take $U \in \mathcal{O}$, I have two problems : If $\mathcal{U} \in J(U)$, why do I ...
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Recall that sites are Morita-equivalent in the sense of 1-topos theory if they have equivalent 1-toposes of sheaves. Question. Let $(\mathcal{C}, \mathsf{J})$ and $(\mathcal{D}, \mathsf{K})$ be sites, ...
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The theory of categories equipped with various limits, colimits and even exponentials can be written as an essentially/generalized algebraic theory. This means we automatically get various notions of ...
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The article on nLab contains numerous interesting examples originating from "geometry defined by analysis." That same article includes methods for constructing new ones in the "examples&...
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There are several well-known theorems constructing new Grothendieck toposes from a given one $E$: The slice category $E_{/c}$ The category of coalgebras of a left-exact comonad The category of ...
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Let $C$ be a subcanonical superextensive site equipped with a certain notion of an open (étale) embedding $\mathrm{Open} \subset \mathrm{Mor}~C$, which generates the site structure in the usual way. ...
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Let $D < C$ be a small full subcategory, then we have the nerve-realization adjunction $$|-| : \operatorname{PSh}(D) \leftrightarrows C : N$$ Under what conditions is this adjunction idempotent? ...
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This question was prompted by this recent answer by aws on a question by Gro-Tsen. As that answer recalls, two PCA’s (partial combinatory algebras) are equivalent via applicative morphisms precisely ...
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I am looking for a topos that describes realizability by Turing machines with access to a “variable oracle”. I think the construction I want is this. Start with Baire space $\mathcal{N} := \mathbb{N}^...
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An elementary topos can be defined as a category with finite limits, exponentials, and a subobject classifier. These notions do not make a size restriction on the collections of morphisms. So in my ...
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The working mathematician and layperson's reasons for caring about constructive foundations are well-documented. Topos theory, in particular, gives an abundance of examples in nature of settings where ...
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A total category is a locally small category $\mathcal{C}$ such that the yoneda embedding $\mathcal{C} \to \textsf{Psh}(\mathcal{C})$ has a left adjoint $L$. It is totally distributive if there is a ...
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This is a question about categories internal to the category of locales, $\mathbf{Loc}$. An étale complete localic groupoid $\mathbb{G}$ gives rise to a localic category, $\gamma \mathbb{G}$, by ...
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This question is about the generalization of Grothendieck Galois theory to topoi. I understand that there is some nuance about the correct definition of Galois topos. But in his paper "On the ...
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I apologize in advance if this is a rather elementary question; I am a physicist and I am still learning the basics of higher topos theory. An $(\infty, 1)$-topos $H$ is morally a category of $(\infty,...
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Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams. For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X)^...
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In constructive analysis, I'm looking at principles which follow both when there exists at least one discontinuous function from $\mathbb{R}$ to $\mathbb{R}$ (equivalent to WLPO i.e. $x > 0$ or $x \...
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I've been reading through Greenblatt's Topoi and while I'm still definitely over my head I'm starting to get a feel for some of the concepts at play there. I see the definitions of $\mathbb{R}_c$ and $...
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Let $C$ be a small (1-)category. There is always a poset $D$ and a functor $p : D \to C$ such that: $p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
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The question is basically "do we really have a good way to talk about large categories internally in an elementary topos?" An internal small category in a topos $E$ is just a category object ...
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I have a question on how to calculate a topos of sheaves on an internal site. Let $F$ be the category of finite sets and functions so that the topos ${\widehat{F}}$ of presheaves on $F$ classifies ...
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The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
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Working in Quine's $\sf NFU$, with urelements being at least as many as sets. Formally the latter is: $|Ur| \geq |Set|$. Where $Ur$ is the set of all urelements and $Set$ is the set of all sets. We ...
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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 ...
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In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?. In the topos of simplicial sets, the subobject ...
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I'm stuck on the following. Let $\mathcal{E}$ be a category satisfying the Giraud axioms: namely $\mathcal{E}$ is locally presentable, has universal colimits, has disjoint coproducts, and has ...
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Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this ...
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Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the ...
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This question is inspired by Homotopy type theory, but I believe it can be thought about also in other constructive foundations. In HoTT the question could be stated as follows: Given a definition of ...
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In Lawvere's Open problems in topos theory; quotient topoi are treated as connected geometric morphisms of Grothendieck topoi. Is there a good reference for where these come from? Is there any sense ...
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Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call ...
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Let $\mathcal{C}$ have products, $A, B \in \mathcal{C}_0$, ${\times}\colon \mathcal{C}^2\to \mathcal{C}$ be the functor that sends objects to their product. Then the induced product ${\boxtimes}\colon ...
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