nLab Sheaves in Geometry and Logic

This entry collects hyperlinks related to the textbook

• Saunders Mac Lane, Ieke Moerdijk:

  
  

  Sheaves in Geometry and Logic

  A first introduction to topos theory

  
  

  Springer (1992)

  doi:10.1007/978-1-4612-0927-0

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

For a similar link lists see also

• Categories and Sheaves

• Sketches of an Elephant

Categorical Preliminaries

I Categories of Functors

• presheaf

• category of presheaves

II Sheaves of Sets

1. Sheaves

2. Sieves and Sheaves

• category of open subsets

• sieve

• sheaf

• global section/constant sheaf

3. Sheaves and Manifolds

4. Bundles

• bundle

5. Sheaves and Cross-Sections

6. Sheaves as Étale spaces

• etale space

7. Sheaves with algebraic structure

• internalization

• group object

8. Sheaves are Typical

9. Inverse Image Sheaf

III Grothendieck Topologies and Sheaves

• presheaf

• site

  • coverage

  • basis for a topology

  • Grothendieck topology

  • sheaf

    • sheafification

    • direct image

    • inverse image

    • restriction and extension of sheaves

    • stalk

  • category of sheaves

• Grothendieck topos

IV First Properties of Elementary Topoi

1. Definition of a topos

• subobject classifier

  • subobject

• topos

  • Grothendieck topos

  • well-pointed topos

  • Boolean topos

  • pretopos

  • universe in a topos

• point of a topos

• fundamental group of a topos

• dependent product

2. The construction of exponentials

• exponential object

3. Direct image

• direct image

4. Monads and Beck’s theorem

• monad

• Eilenberg-Moore category

• monadic functor

• monadicity theorem

5. The construction of colimits

• quotient type

6. Factorization and images

• image

7. The slice category as a topos

• over-topos

8. Lattice and Heyting algebra objects in a topos

• lattice

• Heyting algebra

9. The Beck-Chevalley condition

• Beck-Chevalley condition

10. Injective objects

• injective objects

V Basic Constructions of Topoi

1. Lawvere-Tierney topologies

• Lawvere-Tierney topology

2. Sheaves

• dense monomorphism

• sheaf

3. The associated sheaf functor

• sheafification

4. Lawvere-Tierney subsumes Grothendieck

• Grothendieck topology

5. Internal versus external

6. Group actions

• group, action

• group object, action object

• permutation representation

7. Category actions

8. The topos of coalgebras

• comonad

• topos of coalgebras

9. The filter-quotient construction

VI Topoi and Logic

• ETCS

• continuum hypothesis

• logic

• foundations and logic

  • internal logic

    • intuitionistic logic

    • classical logic

    • paraconsistent logic

    • coherent logic

• Boolean algebra

• Heyting algebra

VII Geometric Morphisms

VII 1. Geometric Morphisms and Basic Examples

• geometric morphism

  • direct image, inverse image

Examples:

• continuous function

• base change geometric morphism

• sheafification, geometric embedding

• global section geometric morphism

VII 2. Tensor products

• Kan extension

VII 3. Group actions

• group object

VII 4. Embeddings and surjections

• geometric embedding

• geometric surjection

• geometric surjection/embedding factorization

VII 5. Points

• point of a topos

• stalk

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

• morphism of sites

• covering lifting property

VIII Classifying Topoi

• classifying topos

IX Localic Topoi

• locale

• localic topos

Geometric Logic and Classifying Topoi

• classifying topos

Appendix: Sites for Topoi

• Giraud's theorem

category: reference