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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Categories in which isomorphism of stalks does not imply isomorphism of sheaves
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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Does this weak omniscience principle have a name?
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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Can the real numbers be constructed as/from a Hom-object in a topos?
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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Covering categories with posets
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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Large "internal" categories and "finite" products
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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Explicit description of a topos of sheaves on an internal boolean algebra
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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Merging single-sorted and multi-sorted theories
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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Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?
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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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 ...
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Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
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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Giraud's axioms imply balanced
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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What topos-theoretic construction lies behind the “symmetric model” construction (used to refute AC) in Set Theory?
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, origin and motivation for the terminology
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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Grothendieck topoi as a constructive property
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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Quotient topoi as quotient objects
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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