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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Showing that the sheaf-functor $\epsilon: \tilde{\sf C} \to \tilde{\tilde{\sf C}}$ is an equivalence

Let $(\mathsf C,J)$ be a site. Then we have the category of sheaves $\tilde{\mathsf C}$ and the category $\tilde{\tilde{\sf C}}$ of sheaves over $\tilde{\sf C}$ (both considered with the canonical ...
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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 ...
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880 views

Fuzzy logic and topos theory

Why doesn't one develop fuzzy logic by extending topos theory, by simply extending the subobject classifier $\Omega$ to the unit interval [0,1]? Have people done that?
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Explicit construction of a initial object in a topos

Let $\mathcal E$ be a topos as in Mac Lane and Moerdijk. A initial object in $\mathcal E$ can be obtained as the domain of the equalizer of the morphisms $P!,\epsilon P1:P1\to P^31$, where $1$ is a ...
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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 ...
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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 $M$...
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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 ...
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749 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
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Could there be an “$n$-th root” of the category $\mathsf{Set}$?

Here is a thought experiment: Suppose we did not know what sets and functions are. The general idea of a topos is, that it somehow serves as a foundation for mathematics. So let there be an alternate ...
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Applications of category theory and topoi/topos theory in reality

I am an amateur mathematician with an interest in the subjects named in the title. I have recently come to understand that my B.A. in math gives me absolutely no qualification at all in the Swedish ...
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The dense topology

The definition of the dense topology confuses me. If $C$ is a category and $X \in C$, a sieve $S$ on $X$ is a covering for the dense topology iff for every $f : Y \to X$ there is some morphism $g : Z \...
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Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?

Here $G$-sets denote the category of sets which have a left $G$-action. So the question is whether a functor $F \colon \text{$G$-sets} \to \text{$H$-sets}$ implies that we have an isomorphism of ...
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A Question on a claim regarding the notion of “space” in “Indiscrete Thoughts”

I'm reading Gian-Carlo Rota's book "Indiscrete Thoughts". In page 220 I came across a strange quotation with very few explanations: We thought that the generalizations of the notion of space had ...
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Quantificators vs pullbacks

Let $\mathscr C$ be a cartesian category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$. Consider the following pullback square: $\require{AMScd} \begin{CD} P @>{\bar f}...
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Grothendieck Topology on the Category of Elements

We are given a site $(C, J)$ for a small category $C$ and a Grothendieck topology $J$. If $F\in Sh(C, J)$, we take the natural topology $J_F$ on its category of elements $el(F)$ induced by $J$. I am ...
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Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
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Equivalent definitions of regular categories?

maybe this is a stupid question, but I could not solve it after some time of meditation. There are four different notions of regular categories: 1) A cartesian category with coequalizers of kernel ...
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Converse to pullback pasting and local diffeomorphisms

The nlab page on local diffeomorphisms gives the following two equivalent conditions for a smoonth function to be a local diffeomorphism. The derivative is an isomorphism of tangent spaces $df:T_xX\...
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Is there a characterization of coverings in subcanonical pretopologies?

Let $\mathcal C$ be a category. A sieve for $\mathcal C$ is called strictly universally epimorphic if it is one of the covering sieves for the canonical topology on $\mathcal C$. SGA4 gives the ...
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Commutation of pullbacks with disjoint unions in sites?

From what I understand, in superextensive sites disjoint unions commute with pullbacks. I know this holds in every topos sense pullback has a right adjoint, and I have heard something along the lines ...