Pseudofunctors and simplicial categories Nick Gurski Case Western Reserve University Pseudofunctors and simplicial categories Overview of the problem Enrichment through variation Nick Gurski Our higher dimensional Case Western Reserve University version Category Theory Octoberfest October 26, 2019 1 / 19
Pseudofunctors This talk has two components and simplicial categories 1. I want tell you about some joint work with Daniel Nick Gurski Case Western Sch¨ appi. Our goal was to construct some explicit Reserve University pseudofunctors from simplicially enriched categories in Overview of the problem the style of Gordon-Power’s Enrichment through Enrichment variation . through variation Our higher dimensional version 2 / 19
Pseudofunctors This talk has two components and simplicial categories 1. I want tell you about some joint work with Daniel Nick Gurski Case Western Sch¨ appi. Our goal was to construct some explicit Reserve University pseudofunctors from simplicially enriched categories in Overview of the problem the style of Gordon-Power’s Enrichment through Enrichment variation . through variation 2. The second implicit component of this talk is an Our higher dimensional invitation for someone else to do this better. version 2 / 19
Pseudofunctors This talk has two components and simplicial categories 1. I want tell you about some joint work with Daniel Nick Gurski Case Western Sch¨ appi. Our goal was to construct some explicit Reserve University pseudofunctors from simplicially enriched categories in Overview of the problem the style of Gordon-Power’s Enrichment through Enrichment variation . through variation 2. The second implicit component of this talk is an Our higher dimensional invitation for someone else to do this better. version ◮ Variable base of enrichment 2 / 19
Pseudofunctors This talk has two components and simplicial categories 1. I want tell you about some joint work with Daniel Nick Gurski Case Western Sch¨ appi. Our goal was to construct some explicit Reserve University pseudofunctors from simplicially enriched categories in Overview of the problem the style of Gordon-Power’s Enrichment through Enrichment variation . through variation 2. The second implicit component of this talk is an Our higher dimensional invitation for someone else to do this better. version ◮ Variable base of enrichment ◮ Variable strength of enrichment 2 / 19
Pseudofunctors This talk has two components and simplicial categories 1. I want tell you about some joint work with Daniel Nick Gurski Case Western Sch¨ appi. Our goal was to construct some explicit Reserve University pseudofunctors from simplicially enriched categories in Overview of the problem the style of Gordon-Power’s Enrichment through Enrichment variation . through variation 2. The second implicit component of this talk is an Our higher dimensional invitation for someone else to do this better. version ◮ Variable base of enrichment ◮ Variable strength of enrichment ◮ But in a usable format 2 / 19
Pseudofunctors Outline and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Overview of the problem Enrichment through variation Our higher dimensional version Enrichment through variation Our higher dimensional version 3 / 19
Pseudofunctors Outline and simplicial categories Nick Gurski Case Western Reserve University Overview of the problem Overview of the problem Enrichment through variation Our higher dimensional version Enrichment through variation Our higher dimensional version 4 / 19
Pseudofunctors Setup and simplicial categories Let E be some nice model category whose objects we think Nick Gurski Case Western of as higher categories. Reserve University Overview of the problem Enrichment through variation Our higher dimensional version 5 / 19
Pseudofunctors Setup and simplicial categories Let E be some nice model category whose objects we think Nick Gurski Case Western of as higher categories. Reserve University ◮ Simplicial sets with Joyal model structure Overview of the problem ◮ Cat with the canonical model structure Enrichment through variation Our higher dimensional version 5 / 19
Pseudofunctors Setup and simplicial categories Let E be some nice model category whose objects we think Nick Gurski Case Western of as higher categories. Reserve University ◮ Simplicial sets with Joyal model structure Overview of the problem ◮ Cat with the canonical model structure Enrichment through variation We can study Our higher ◮ s E : simplicial objects in E , dimensional version ◮ qCat ( E ) : internal quasicategories in E , and ◮ Segal ( E ) : internal Segal categories in E . 5 / 19
Pseudofunctors Application and simplicial categories We can write down a functor Nick Gurski Case Western Reserve University qCat ( E ) → Cat Overview of the problem by Enrichment through variation A �→ Ho( s E /A ) . Our higher dimensional version 6 / 19
Pseudofunctors Application and simplicial categories We can write down a functor Nick Gurski Case Western Reserve University qCat ( E ) → Cat Overview of the problem by Enrichment through variation A �→ Ho( s E /A ) . Our higher dimensional Question : Can we extend this to a functor between higher version categories? 6 / 19
Pseudofunctors Application and simplicial categories We can write down a functor Nick Gurski Case Western Reserve University qCat ( E ) → Cat Overview of the problem by Enrichment through variation A �→ Ho( s E /A ) . Our higher dimensional Question : Can we extend this to a functor between higher version categories? Higher dimensional versions: ◮ Cat is a 2-category* ◮ qCat ( E ) is a simplicially enriched category 6 / 19
Pseudofunctors Application and simplicial categories We can write down a functor Nick Gurski Case Western Reserve University qCat ( E ) → Cat Overview of the problem by Enrichment through variation A �→ Ho( s E /A ) . Our higher dimensional Question : Can we extend this to a functor between higher version categories? Higher dimensional versions: ◮ Cat is a 2-category* ◮ qCat ( E ) is a simplicially enriched category Conclusion : we are looking for functors from a simplicially enriched category to a 2-category. 6 / 19
Pseudofunctors Graphs first and simplicial categories Recall the adjunction τ 1 ⊣ N Nick Gurski Case Western ◮ N : sSet → Cat , Reserve University ◮ τ 1 : Cat → sSet . Overview of the problem Enrichment through variation Our higher dimensional version 7 / 19
Pseudofunctors Graphs first and simplicial categories Recall the adjunction τ 1 ⊣ N Nick Gurski Case Western ◮ N : sSet → Cat , Reserve University ◮ τ 1 : Cat → sSet . Overview of the problem Lemma Enrichment through variation For any adjunction F ⊣ U with F : A ⇄ B and any category Our higher dimensional D , there is an induced adjunction version F ∗ ⊣ U ∗ with F ∗ : [ D , A ] ⇄ [ D , B ] : U ∗ . 7 / 19
Pseudofunctors Graphs first and simplicial categories Recall the adjunction τ 1 ⊣ N Nick Gurski Case Western ◮ N : sSet → Cat , Reserve University ◮ τ 1 : Cat → sSet . Overview of the problem Lemma Enrichment through variation For any adjunction F ⊣ U with F : A ⇄ B and any category Our higher dimensional D , there is an induced adjunction version F ∗ ⊣ U ∗ with F ∗ : [ D , A ] ⇄ [ D , B ] : U ∗ . Apply with D = m ⇒ o to the adjunction above to get ( τ 1 ) ∗ ⊣ N ∗ with ( τ 1 ) ∗ : Graph ( sSet ) ⇄ Graph ( Cat ) : N ∗ . 7 / 19
Pseudofunctors From graphs to enriched categories and simplicial categories Our lemma constructing F ∗ ⊣ U ∗ for graphs can be extended. Nick Gurski Case Western Reserve University Proposition Overview of the 1. If P : A → B is a lax monoidal functor between problem monoidal categories, then it induces a functor Enrichment through variation Our higher P : Mon ( A ) → Mon ( B ) . dimensional version 8 / 19
Pseudofunctors From graphs to enriched categories and simplicial categories Our lemma constructing F ∗ ⊣ U ∗ for graphs can be extended. Nick Gurski Case Western Reserve University Proposition Overview of the 1. If P : A → B is a lax monoidal functor between problem monoidal categories, then it induces a functor Enrichment through variation Our higher P : Mon ( A ) → Mon ( B ) . dimensional version 2. Under the same hypotheses, P induces a functor P ∗ : A - Cat → B - Cat by applying P to the hom-objects. 8 / 19
Pseudofunctors From graphs to enriched categories and simplicial categories Our lemma constructing F ∗ ⊣ U ∗ for graphs can be extended. Nick Gurski Case Western Reserve University Proposition Overview of the 1. If P : A → B is a lax monoidal functor between problem monoidal categories, then it induces a functor Enrichment through variation P : Mon ( A ) → Mon ( B ) . Our higher dimensional version 2. Under the same hypotheses, P induces a functor P ∗ : A - Cat → B - Cat by applying P to the hom-objects. 3. If F ⊣ U is a monoidal adjunction, it induces an adjunction F ∗ : A - Cat ⇄ B - Cat : U ∗ . 8 / 19
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