# Questions tagged [topos-theory]

A topos (plural topoi, toposes) is a category that behaves like the category of sheaves of sets on a topological space. Topos theory consists of the study of Grothendieck topoi, used in algebraic geometry, and the study of elementary topoi, used in logic.

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### What concretely does it mean to "do mathematics in a topos X"?

I don't know if this is the right kind of question, but I've heard people say this phrase "do mathematics in a topos", and the idea that Set is the topos people usually work in, but only one ...
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### Is the category of undirected graphs an infinity-pretopos?

Theorem C2.2.8 in "Sketches of an Elephant" by P.T. Johnstone states that an $\infty$-pretopos with a separating set of objects is a topos. However, the category of undirected graphs seem to ...
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### Is every Heyting algebra the subobject classifier of some topos?

Let $\mathcal{E}$ be an elementary topos. I know that the subobject classifier $\Omega$ of the topos is always a Heyting algebra. I’m interested in versions of the converse — given any (let's say ...
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### About F.Lawvere article "AN ELEMENTARY THEORY OF THE CATEGORY OF SETS"

reference articles: [L] AN ELEMENTARY THEORY OF THE CATEGORY OF SETS (LONG VERSION) WITH COMMENTARY, F. WILLIAM LAWVERE (http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf) [T] Topos theory, P....
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### Show that the smallest such Grothendieck topology is the open cover topology

This is an exercise in Maclane-Moerdijk (III.6.(b)). Let $\mathbf{C}$ be a small category, and $S_\alpha$ any collection of sieves on objects $C_\alpha$. It is easy to show that there is a smallest ...
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### What is the motivation of topos theory?

What is the motivation of topos theory? https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k&index=1 So from my understanding, the motivation of a topos is to ...
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### How should we really think about subobjects in a topos?

Scenesetting Some define a subobject of $X$ as a monic $s \colon S \rightarrowtail X.$ For familiar reasons, though, most officially define a subobject of $X$ as an equivalence class $[s]$ of such ...
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### Does an internal topos object of a topos induce an "iterated topos"?

From the internal language of a topos, especially the dependently typed version, it seems reasonable to be able to define the notion of an internal topos object. For example, you would start with an ...
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### Zorn's Lemma in Topos Theory

I am reading the book "Sheaves in Geometry and Logic". When constructing the topos of sets, there is a proposition (proposition 8 of the "Topos of sets" subchapter) that uses Zorn'...
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### Can we sheaf-theoretically force a violation of the continuum hypothesis in a (nice) topos which is *bicomplete*?

$\newcommand{\p}{\mathcal{P}}$I recently dug through the exercises and details in Mac Lane and Moerdijk's book "Sheaves in Geometry and Logic" which concern themselves with (a baby version ...
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### Are there Kripke-Joyal semantics for terms which involve arrows of the topos (rather than purely variables and connectives)?

I follow Mac Lane and Moerdijk's (slightly informal) presentation of the Mitchell-Benabou language. There, they outline several recursive rules to describe what the terms of the language are and ...
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### Explicit geometric machinery to check the truth of FOL formula

It is standard to use the semantics of topological spaces for checking the truth value of propositional formula in the absence of excluded middle. An explicit description is written down here, for ...
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### Examples of geometric theory without classical models

I encountered a post on locales and geometric theory here https://grossack.site/2022/05/22/locale-basics.html In about the middle of this blog, the author gives a geometric theory defining a function ...
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### Elementary proofs of distributive and other properties of $\land$, $\lor$, $\neg$ in topos?

Take the 'logical' arrows $\wedge \colon \Omega \times \Omega \to \Omega$, $\lor \colon \Omega \times \Omega \to \Omega$, $\neg \colon \Omega \to \Omega$, defined by the usual pullback squares, as in ...
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### proposition 6.22 in Awodey's category theory textbook

Proposition 6.22 reads: For every CCC (Cartesian closed category) $C$, there is a poset $I$ and a functor, $$y: C \to \textrm{Sets}^I,$$ that is injective on both objects and arrows and preserves CCC ...
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### Zariski opens of an affine space and (po)sites

There is a very neat presentation of the Zariski open sets in $\operatorname{Spec}(R)$: it is freely generated (under arbitrary unions and finite intersections, which distribute over each other) by ...
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### 0-connective objects in an $\infty$-topos

I am trying to understand Marc Hoyois's proof of the Van Kampen theorem in an $\infty$-topos given as Lemma 6 in : https://ncatlab.org/nlab/files/Hoyois_PlusConstruction.pdf In the course of the proof,...
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### Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?

I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the ...
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### Are the "intramural" and "extramural" definitions of "set" from category theory equivalent?

Note: This question takes a pluralistic / "multiverse" view of sets (including "classes" and "collections") and set theories. My understanding (from reading many nLab ...
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### Ordinary functor being E-indexed

I am learning parts of topos theory and it feels like I am missing something regarding indexed functors. This comes from the definition of a locally connected geometric morphism $f : E \rightarrow F$ ...
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### Doubt in proof showing in topoi, equivalence relations are effective

This is proposition 1.4 (pg. 30) in van Oosten's lecture notes on Topos Theory. Prop: In a topos, every equivalence relation is effective, i.e. a kernel pair. Following is the first half of the ...
I came across this definition while reading Jaap van Oosten's Topos Theory lecture notes (pg 29, def 1.3). In a category with finite limits, an equivalence relation on an object $X$ is a subobject $R$ ...