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Results tagged with topos-theory
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user 471475
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,...
4
votes
1
answer
182
views
Group objects via diagrams or generalized elements — Kripke–Joyal?
The notion of a group object $G$ in a category with finite products can either be defined with a few commutative diagrams or via requiring that each hom set $\hom(X,G)$ is a group. There is a theorem …
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12
votes
1
answer
410
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Definition of "classifying topos"
Is there a geometric theory $T$ and a Grothendieck topos $\mathcal E$ such that (2) holds but (1) doesn't:
$\mathcal E$ 2-represents the 2-functor
$$\mathbf{GrothTop}\to\mathbf{Cat}$$
which sends a G …
- 723
23
votes
2
answers
2k
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An extension of the Galois theory of Grothendieck
This question is about Joyal and Tierney's famous An extension of the Galois theory of Grothendieck. One of the main results states (see the MathSciNet review by Peter Johnstone):
Joyal and Tierney's …
- 723
7
votes
1
answer
271
views
Direct and inverse image terminology
Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and …
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16
votes
2
answers
1k
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Major applications of the internal language of toposes
What are the major applications of the internal language of toposes?
Here are a few applications I know:
Mulvey's proof of the Serre–Swan theorem in which he interprets the intuitionistically valid r …