nLab
2-topos

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Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

Contents

Idea

The notion of 2-topos is the generalization of the notion of topos from category theory to the higher category theory of 2-categories.

There are multiple conceivable such generalizations, depending in particular on whether one tries to generalize the notion of Grothendieck topos or of elementary topos, and in the latter case what axioms one chooses to take as the basis for generalization.

In contrast, (2,1)-toposes are much better understood.

A Grothendieck 2-topos is a 2-category of 2-sheaves over a 2-site.

A Grothendieck (2,1)-topos is a (2,1)-category of (2,1)-sheaves over a (2,1)-site.

See also higher topos theory.

Examples

The archetypical 2-topos is Cat.

References

Revised on September 11, 2011 18:16:30 by Urs Schreiber (82.113.99.24)
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