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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.

Topos theory has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kind of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic.

Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories.

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What does it take to divide by $2$?

Theorem 1 [ZFC, classical logic]: If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$. That's because the axiom of choice allows for the definition of cardinality $|A|$ of any set $A$, and for…
Hanno
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Does "cheap nonstandard analysis" take place in a topos?

Terence Tao's A cheap version of nonstandard analysis describes a way to do analysis halfway between ordinary analysis and nonstandard analysis which, if I'm not mistaken, cashes out to working in the following category $C$ instead of $\text{Set}$.…
Qiaochu Yuan
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What does a proof in an internal logic actually look like?

The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. However, I can't find a single example on the nLab…
Qiaochu Yuan
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How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$ determined by $\mathbf{Set}_M$ as a category up…
Zhen Lin
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What is Mazzola's "Topos of Music" about?

Disclaimers: I am neither a musician, nor I want to discredit Mazzola's work. Corollary of the first point: please use a plain style, without technical terms in the area of Music Theory. Corollary of the second: don't take my disbelief in Mazzola's…
fosco
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How to find exponential objects and subobject classifiers in a given category

In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject classifier, and to show that it has all finite…
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How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the entities in the definition of an elementary topos come…
Arrow
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Revisiting "What is Mazzola's "Topos of Music" about?"

This question What is Mazzola's "Topos of Music" about? has already been asked, but I am dissatisfied with the response for several reasons and would like Math SE to revisit it. For starters, no one answered the question in the title. Remark Topos…
Stan Shunpike
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What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). However, vaguely speaking by a logic I mean a…
user40276
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Elementary topoi have initial objects, why?

An elementary topos is a category that: has finite limits is cartesian closed has a subobject classifier and one can show (with quite a bit of effort) that it has finite colimits. Is there, starting from the axioms, a quick and easy way to see…
Stefan Perko
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Is there a category whose internal logic is paraconsistent?

The internal language of topoi is higher-order typed intuitionistic logic. Now according to wikipedia, the dual of intuitionistic logic, in some sense is paraconsistent. They say Intuitionistic logic allows $A ∨ ¬A$ not to be equivalent to true,…
Mozibur Ullah
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What are topoi?

I have been hearing a lot about the concept of "topos". I asked a friend of mine in the know and he said that topoi are a generalization of sheaves on a topological space. In particular, topoi were usefull when an actual topology was not available.…
WhoKnowsMyName
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When does Sheafification commute with direct image?

Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since sheafification is a left adjoint and pushforward is a…
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Importance of 'smallness' in a category, and functor categories

I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I continually fail to see why this is: I can see why…
Spyam
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What are the algebras of the double powerset monad?

Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image map $f^* : \mathscr{P}(Y) \to \mathscr{P}(X)$. By…
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