limits and colimits

1-Categorical

• limit and colimit

  • limits and colimits by example

  • commutativity of limits and colimits

  • small limit

  • filtered colimit

    • directed colimit

      • sequential colimit

  • sifted colimit

  • connected limit, wide pullback

  • preserved limit, reflected limit, created limit

  • product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum

  • finite limit

    • exact functor

• Kan extension

  • Yoneda extension

• weighted limit

• end and coend

• fibered limit

2-Categorical

• 2-limit

  • inserter

  • isoinserter

  • equifier

  • inverter

  • PIE-limit

• 2-pullback, comma object

(∞,1)-Categorical

• (∞,1)-limit

  • (∞,1)-pullback

    • fiber sequence

Model-categorical

• homotopy Kan extension

• homotopy limit

  • homotopy product

  • homotopy equalizer

  • homotopy fiber

  • mapping cone

  • homotopy pullback

  • homotopy totalization

  • homotopy end

• homotopy colimit

  • homotopy coproduct

  • homotopy coequalizer

  • homotopy cofiber

  • mapping cocone

  • homotopy pushout

  • homotopy realization

  • homotopy coend

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

• Introduction to Basic Homotopy Theory

• Introduction to Abstract Homotopy Theory

• geometry of physics – homotopy types

Definitions

• homotopy, higher homotopy

• homotopy type

• Pi-algebra, spherical object and Pi(A)-algebra

• homotopy coherent category theory

  • homotopical category

    • model category

    • category of fibrant objects, cofibration category

    • Waldhausen category

  • homotopy category

    • Ho(Top)

• (∞,1)-category

  • homotopy category of an (∞,1)-category

Paths and cylinders

• left homotopy

  • cylinder object

  • mapping cone

• right homotopy

  • path object

  • mapping cocone

  • universal bundle

• interval object

  • homotopy localization

  • infinitesimal interval object

Homotopy groups

• homotopy group

  • fundamental group

    • fundamental group of a topos

  • Brown-Grossman homotopy group

  • categorical homotopy groups in an (∞,1)-topos

  • geometric homotopy groups in an (∞,1)-topos

• fundamental ∞-groupoid

  • fundamental groupoid

    • path groupoid

  • fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos

  • fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos

• fundamental (∞,1)-category

  • fundamental category

Basic facts

• fundamental group of the circle is the integers

Theorems

• fundamental theorem of covering spaces

• Freudenthal suspension theorem

• Blakers-Massey theorem

• higher homotopy van Kampen theorem

• nerve theorem

• Whitehead's theorem

• Hurewicz theorem

• Galois theory

• homotopy hypothesis-theorem

model category, model $\infty$-category

Definitions

• category with weak equivalences

  (relative category, homotopical category)

• fibration, cofibration

• weak factorization system

• resolution

• cell complex

• small object argument

• homotopy

• homotopy category$\;$of a model category

Morphisms

• Quillen adjunction

  • Quillen equivalence

  • Quillen bifunctor

  • derived functor

Universal constructions

• homotopy Kan extension

• homotopy limit/homotopy colimit

  homotopy weighted (co)limit

  homotopy (co)end

• Bousfield-Kan map

Refinements

• monoidal model category

  • monoidal Quillen adjunction

• enriched model category

  • enriched Quillen adjunction

• monoidal enriched model category

• simplicial model category

  • simplicial Quillen adjunction

  • simplicial monoidal model category

• cofibrantly generated model category

  • combinatorial model category

  • cellular model category

• algebraic model category

• compactly generated model category

• proper model category

• cartesian closed model category, locally cartesian closed model category

• stable model category

Producing new model structures

• on functor categories (global)

  • Reedy model structure

• on slice categories

• Bousfield localization

• transferred model structure

  • on algebraic fibrant objects

• Grothendieck construction for model categories

Presentation of $(\infty,1)$-categories

• (∞,1)-category

• simplicial localization

• (∞,1)-categorical hom-space

• presentable (∞,1)-category

Model structures

• Cisinski model structure

for $\infty$-groupoids

for ∞-groupoids

• on topological spaces

  • classical model structure

  • on compactly generated spaces

  • on Delta-generated spaces

  • on diffeological spaces

  • Strøm model structure

• Thomason model structure

• model structure on presheaves over a test category

• on simplicial sets, on semi-simplicial sets

  • classical model structure

  • constructive model structure

  • for right/left fibrations

• model structure on simplicial groupoids

• on cubical sets

• on strict ∞-groupoids, on groupoids

• on chain complexes/model structure on cosimplicial abelian groups

  related by the Dold-Kan correspondence

• model structure on cosimplicial simplicial sets

for equivariant $\infty$-groupoids

• fine model structure on topological G-spaces

• coarse model structure on topological G-spaces

  (Borel model structure)

for rational $\infty$-groupoids

• model structure on dgc-algebras

for rational equivariant $\infty$-groupoids

• model structure on equivariant chain complexes

• model structure on equivariant dgc-algebras

for $n$-groupoids

• for n-groupoids/for n-types

• for 1-groupoids

for $\infty$-groups

• model structure on simplicial groups

• model structure on reduced simplicial sets

for $\infty$-algebras

general $\infty$-algebras

• on monoids

• on simplicial T-algebras, on homotopy T-algebras

• on algebas over a monad

• on algebras over an operad,

  on modules over an algebra over an operad

specific $\infty$-algebras

• model structure on differential-graded commutative algebras

• model structure on differential graded-commutative superalgebras

• on dg-algebras over an operad

  • on dg-algebras and on on simplicial rings/on cosimplicial rings

    related by the monoidal Dold-Kan correspondence

  • for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras

• model structure on dg-modules

for stable/spectrum objects

• model structure on spectra

• model structure on ring spectra

• model structure on parameterized spectra

• model structure on presheaves of spectra

for $(\infty,1)$-categories

• on categories with weak equivalences

• Joyal model for quasi-categories (and its cubical version)

• on sSet-categories

• for complete Segal spaces

• for Cartesian fibrations

for stable $(\infty,1)$-categories

• on dg-categories

for $(\infty,1)$-operads

• on operads, for Segal operads

  on algebras over an operad,

  on modules over an algebra over an operad

• on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations

for $(n,r)$-categories

• for (n,r)-categories as ∞-spaces

• for weak ∞-categories as weak complicial sets

• on cellular sets

• on higher categories in general

• on strict ∞-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

• on homotopical presheaves

  • on simplicial presheaves

    global model structure/Cech model structure/local model structure

    on simplicial sheaves

    on presheaves of simplicial groupoids

    on sSet-enriched presheaves

• model structure for (2,1)-sheaves/for stacks