topos theory

• Toposes

Background

• category theory

  • category

  • functor

Toposes

• (0,1)-topos, Heyting algebra, locale

• pretopos

• topos

  • Grothendieck topos

    • category of presheaves

      • presheaf

      • representable presheaf

    • category of sheaves

      • site

        • sieve

        • coverage, pretopology, topology

      • sheaf

      • sheafification

• quasitopos

• base topos, indexed topos

Internal Logic

• categorical semantics

• internal logic

  • subobject classifier

  • natural numbers object

Topos morphisms

• logical morphism

• geometric morphism

  • direct image/inverse image

  • global sections

  • geometric embedding

  • surjective geometric morphism

  • essential geometric morphism

    • locally connected geometric morphism

    • connected geometric morphism

    • totally connected geometric morphism

  • étale geometric morphism

  • open geometric morphism

  • proper geometric morphism, compact topos

  • separated geometric morphism, Hausdorff topos

  • local geometric morphism

  • bounded geometric morphism

  • base change

  • localic geometric morphism

  • hyperconnected geometric morphism

  • atomic geometric morphism

Extra stuff, structure, properties

• topological locale

• localic topos

• petit topos/gros topos

• locally connected topos, connected topos, totally connected topos, strongly connected topos

• local topos

• cohesive topos

• classifying topos

• smooth topos

Cohomology and homotopy

• cohomology

• homotopy

• abelian sheaf cohomology

• model structure on simplicial presheaves

In higher category theory

• higher topos theory

• (0,1)-topos

  • (0,1)-site

• 2-topos

  • 2-site

  • 2-sheaf, stack

• (∞,1)-topos

  • (∞,1)-site

  • (∞,1)-sheaf, ∞-stack, derived stack

Theorems

• Diaconescu's theorem

• Barr's theorem

homological algebra

(also nonabelian homological algebra)

Introduction

Context

• additive and abelian categories

  • Ab-enriched category

  • pre-additive category

  • additive category

  • pre-abelian category

  • abelian category

  • Grothendieck category

• abelian sheaves

• semi-abelian category

Basic definitions

• kernel, cokernel

• complex

  • differential

  • homology

• category of chain complexes

  • chain complex

  • chain map

  • chain homotopy

• chain homology and cohomology

  • quasi-isomorphism

  • homological resolution

  • simplicial homology

  • generalized homology

• exact sequence,

  • short exact sequence, long exact sequence, split exact sequence

• injective object, projective object

  • injective resolution, projective resolution

  • flat resolution

Stable homotopy theory notions

• derived category

• triangulated category, enhanced triangulated category

• stable (∞,1)-category

  • stable model category

  • pretriangulated dg-category

  • A-∞-category

• (∞,1)-category of chain complexes

• derived functor, derived functor in homological algebra

  • Tor, Ext

  • homotopy limit, homotopy colimit

  • lim^1 and Milnor sequences

  • fr-code

• abelian sheaf cohomology

Constructions

• double complex

• Koszul-Tate resolution, BRST-BV complex

• spectral sequence

  • spectral sequence of a filtered complex

  • spectral sequence of a double complex

  • Grothendieck spectral sequence

    • Leray spectral sequence

    • Serre spectral sequence

    • Hochschild-Serre spectral sequence

Lemmas

diagram chasing

• 3x3 lemma

• four lemma, five lemma

• snake lemma, connecting homomorphism

• horseshoe lemma

• Baer's criterion

Schanuel's lemma

Homology theories

• singular homology

• cyclic homology

Theorems

• Dold-Kan correspondence / monoidal, operadic

  • Moore complex, Alexander-Whitney map, Eilenberg-Zilber map

• Eilenberg-Zilber theorem

  • cochain on a simplicial set

• universal coefficient theorem

• Künneth theorem