Weak adjoint functor theorems Stephen Lack Macquarie University - PowerPoint PPT Presentation

Weak adjoint functor theorems Stephen Lack Macquarie University CT2019, Edinburgh joint work with John Bourke and Luk´ aˇ s Vokˇ r´ ınek

Adjoint Functor Theorems ur-AFT category B with all limits U : B → A preserves them U has left adjoint

Adjoint Functor Theorems ur-AFT category B with all limits U : B → A preserves them U has left adjoint General AFT category B with small limits U : B → A preserves them Solution Set Condition U has left adjoint

more Adjoint Functor Theorems General AFT (Freyd) Enriched AFT (Kelly) category B with small limits V -category B with small limits U : B → A preserves them U : B → A preserves them SSC SSC U has left adjoint U has left adjoint Weak AFT (Kainen) category B with small products U : B → A preserves them SSC U has weak left adjoint

more Adjoint Functor Theorems General AFT (Freyd) Enriched AFT (Kelly) category B with small limits V -category B with small limits U : B → A preserves them U : B → A preserves them SSC SSC U has left adjoint U has left adjoint Weak AFT (Kainen) category B with small products U : B → A preserves them SSC U has weak left adjoint η A UFA FA Uf ′ ∃ f ′ f UB B

more Adjoint Functor Theorems General AFT (Freyd) Enriched AFT (Kelly) category B with small limits V -category B with small limits U : B → A preserves them U : B → A preserves them SSC SSC U has left adjoint U has left adjoint Weak AFT (Kainen) Very (!) General AFT category B with small products V -category B with � limits U : B → A preserves them U : B → A preserves them SSC SSC U has weak left adjoint U has � left adjoint η A UFA FA Uf ′ ∃ f ′ f UB B

more Adjoint Functor Theorems General AFT (Freyd) Enriched AFT (Kelly) category B with small limits V -category B with small limits U : B → A preserves them U : B → A preserves them SSC SSC U has left adjoint U has left adjoint Weak AFT (Kainen) Very (!) General AFT category B with small products V -category B with � limits U : B → A preserves them U : B → A preserves them SSC SSC U has weak left adjoint U has � left adjoint η A UFA FA Uf ′ ∃ f ′ f UB B

Enriched weakness η A UFA FA surj. B ( FA , B ) A ( A , UB ) Uf ′ ∃ f ′ f UB B ◮ enriched categories have homs C ( C , D ) lying in V ◮ (Lack-Rosicky) “Enriched Weakness” uses class E of morphisms in V to play the role of surjections ◮ V = Set , E = { surjections } gives unenriched weakness ◮ E = { isomorphisms } gives “non-weak weakness”

Enriched weakness η A UFA FA E -map B ( FA , B ) A ( A , UB ) Uf ′ ∃ f ′ f UB B ◮ enriched categories have homs C ( C , D ) lying in V ◮ (Lack-Rosicky) “Enriched Weakness” uses class E of morphisms in V to play the role of surjections ◮ V = Set , E = { surjections } gives unenriched weakness ◮ E = { isomorphisms } gives “non-weak weakness” Very (!) General AFT V -category B with � limits U : B → A preserves them SSC U has E -weak left adjoint

Examples E B ( FA , B ) A ( A , UB ) V E E -weak left adjoint isos left adjoint Set Set surjections weak left adjoint V isos (enriched) left adjoint Cat equivalences left biadjoint

Examples E B ( FA , B ) A ( A , UB ) V E E -weak left adjoint isos left adjoint Set Set surjections weak left adjoint V isos (enriched) left adjoint Cat equivalences left biadjoint Cat surj equivalences (. . . ) left biadjoint

Examples E B ( FA , B ) A ( A , UB ) V E E -weak left adjoint isos left adjoint Set Set surjections weak left adjoint V isos (enriched) left adjoint Cat surj equivalences (. . . ) left biadjoint dual strong deformation retracts sSet Definition A morphism p : X → Y of simplicial sets is dsdr if it is contractible in sSet / Y : ◮ it has a section s ◮ with a homotopy s ◦ p ∼ 1 X ◮ such that induced homotopy p ◦ s ◦ p ∼ p is trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . .

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P .

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization i w I + I J I ∇ of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization i w I + I J I f C ( A , B ) I of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization i w I + I J I ( f g ) h C ( A , B ) C ( A , B ) of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization i w I + I J I ( f = g ) h C ( A , B ) C ( A , B ) of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization i w I + I J I X p ( f g ) h C ( A , B ) C ( A , B ) Y of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization i w I + I J I X p ( f g ) h s C ( A , B ) C ( A , B ) Y Y 1 of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

The setting Let V be a monoidal model category with cofibrant unit I . . . cofibrations I , weak equivalences W , trivial fibrations P . An interval in V is a factorization 1 i w I + I J I X X ≃ p p ( f g ) h s C ( A , B ) C ( A , B ) Y Y 1 of the codiagonal with i ∈ I and w ∈ W . ◮ A morphism A → B in a V -category C is a morphism I → C ( A , B ). ◮ A homotopy between morphisms A → B is a morphism J → C ( A , B ) (for some interval) ◮ A homotopy is trivial if it factorizes through w ◮ A morphism p is dsdr if there exist s with p ◦ s = 1, and homotopy s ◦ p ∼ 1 with p ◦ s ◦ p ∼ p trivial.

Examples (of V ) V I W dsdr morphisms Set all isos isos V all isos isos Set mono all surj inj obj equiv surj equiv Cat sSet mono wk hty equiv (Quillen) dsdr mono wk cat equiv (Joyal) dsdr sSet 2-Cat biequivalences surj, full biequivalences

Examples (of V ) V I W dsdr morphisms Set all isos isos V all isos isos Set mono all surj inj obj equiv surj equiv Cat sSet mono wk hty equiv (Quillen) dsdr mono wk cat equiv (Joyal) dsdr sSet 2-Cat biequivalences surj, full biequivalences In general, for f : X → Y in V : ◮ trivial fibration ⇒ dsdr ( if X , Y cofibrant ) ◮ dsdr ⇒ weak equivalence (if X fibrant or cofibrant)