#
nLab

Sheaves in Geometry and Logic

### Context

#### Topos Theory

**topos theory**

## Background

## Toposes

## Internal Logic

## Topos morphisms

## Cohomology and homotopy

## In higher category theory

## Theorems

This entry collects hyperlinks related to the textbook

on sheaf and topos theory and its application in categorical logic.

For a similar link lists see also

# Contents

## Categorical Preliminaries

## I Categories of Functors

## II Sheaves of Sets

### 1. Sheaves

### 2. Sieves and Sheaves

### 3. Sheaves and Manifolds

### 4. Bundles

### 5. Sheaves and Cross-Sections

### 6. Sheaves as Étale spaces

### 7. Sheaves with algebraic structure

### 8. Sheaves are Typical

### 9. Inverse Image Sheaf

## III Grothendieck Topologies and Sheaves

## IV First Properties of Elementary Topoi

### 1. Definition of a topos

### 2. The construction of exponentials

### 3. Direct image

### 4. Monads and Beck’s theorem

### 5. The construction of colimits

### 6. Factorization and images

### 7. The slice category as a topos

### 8. Lattice and Heyting algebra objects in a topos

### 9. The Beck-Chevalley condition

### 10. Injective objects

## V Basic Constructions of Topoi

### 1. Lawvere-Tierney topologies

### 2. Sheaves

### 3. The associated sheaf functor

### 4. Lawvere-Tierney subsumes Grothendieck

### 5. Internal versus external

### 6. Group actions

### 7. Category actions

### 8. The topos of coalgebras

### 9. The filter-quotient construction

## VI Topoi and Logic

## VII Geometric Morphisms

### VII 1. Geometric Morphisms and Basic Examples

Examples:

### VII 2. Tensor products

### VII 3. Group actions

### VII 4. Embeddings and surjections

### VII 5. Points

### VII 6. Filtering functors

### VII 7. Morphisms into Grothendieck Topoi

### VII 8. Filtering functors into a topos

### VII 9. Geometric morphisms as filtering functors

### VII 10. Morphisms between sites

## VIII Classifying Topoi

## IX Localic Topoi

## Geometric Logic and Classifying Topoi

## Appendix: Sites for Topoi