A proof of the model-independence of (, 1) -category theory joint - PowerPoint PPT Presentation

Emily Riehl Johns Hopkins University A proof of the model-independence of (∞, 1) -category theory joint with Dominic Verity CT2018, Universidade dos Açores

1. What are model-independent foundations? 2. ∞ -cosmoi of (∞, 1) -categories 3. A taste of the formal category theory of (∞, 1) -categories 4. The proof of model-independence of (∞, 1) -category theory Plan Goal: build model-independent foundations of (∞, 1) -category theory

Plan Goal: build model-independent foundations of (∞, 1) -category theory 1. What are model-independent foundations? 2. ∞ -cosmoi of (∞, 1) -categories 3. A taste of the formal category theory of (∞, 1) -categories 4. The proof of model-independence of (∞, 1) -category theory

1 What are model-independent foundations?

• topological categories and relative categories are the simplest to define but do not have enough maps between them quasi-categories (nee. weak Kan complexes) , Rezk spaces (nee. complete Segal spaces) , Segal categories , and (saturated 1-trivial weak) 1-complicial sets each have a homotopically meaningful internal hom. { ⎩ { { ⎨ ⎧ { • Models of (∞, 1) -categories Schematically, an (∞, 1) -category is a category “weakly enriched” over ∞ -groupoids/homotopy types … but this is tricky to make precise. R ezk S egal R el C at T op- C at 1 - C omp q C at

quasi-categories (nee. weak Kan complexes) , Rezk spaces (nee. complete Segal spaces) , Segal categories , and (saturated 1-trivial weak) 1-complicial sets each have a homotopically meaningful internal hom. { ⎩ { { ⎨ { ⎧ • Models of (∞, 1) -categories Schematically, an (∞, 1) -category is a category “weakly enriched” over ∞ -groupoids/homotopy types … but this is tricky to make precise. R ezk S egal R el C at T op- C at 1 - C omp q C at • topological categories and relative categories are the simplest to define but do not have enough maps between them

⎧ • ⎩ { { ⎨ { { Models of (∞, 1) -categories Schematically, an (∞, 1) -category is a category “weakly enriched” over ∞ -groupoids/homotopy types … but this is tricky to make precise. R ezk S egal R el C at T op- C at 1 - C omp q C at • topological categories and relative categories are the simplest to define but do not have enough maps between them quasi-categories (nee. weak Kan complexes) , Rezk spaces (nee. complete Segal spaces) , Segal categories , and (saturated 1-trivial weak) 1-complicial sets each have a homotopically meaningful internal hom.

Two strategies: • work analytically to give categorical definitions and prove theorems using the combinatorics of one model (eg., Joyal, Lurie, Gepner-Haugseng, Cisinski in q at; Kazhdan-Varshavsky, Rasekh in ezk; Simpson in egal) • work synthetically to give categorical definitions and prove theorems in all four models q at, ezk, egal, 1 - omp at once Our method: introduce an ∞ -cosmos to axiomatize common features of the categories q at, ezk, egal, 1 - omp of (∞, 1) -categories. The analytic vs synthetic theory of (∞, 1) -categories Q: How might you develop the category theory of (∞, 1) -categories?

• work synthetically to give categorical definitions and prove theorems in all four models q at, ezk, egal, 1 - omp at once Our method: introduce an ∞ -cosmos to axiomatize common features of the categories q at, ezk, egal, 1 - omp of (∞, 1) -categories. The analytic vs synthetic theory of (∞, 1) -categories Q: How might you develop the category theory of (∞, 1) -categories? Two strategies: • work analytically to give categorical definitions and prove theorems using the combinatorics of one model (eg., Joyal, Lurie, Gepner-Haugseng, Cisinski in q C at; Kazhdan-Varshavsky, Rasekh in R ezk; Simpson in S egal)

Our method: introduce an ∞ -cosmos to axiomatize common features of the categories q at, ezk, egal, 1 - omp of (∞, 1) -categories. The analytic vs synthetic theory of (∞, 1) -categories Q: How might you develop the category theory of (∞, 1) -categories? Two strategies: • work analytically to give categorical definitions and prove theorems using the combinatorics of one model (eg., Joyal, Lurie, Gepner-Haugseng, Cisinski in q C at; Kazhdan-Varshavsky, Rasekh in R ezk; Simpson in S egal) • work synthetically to give categorical definitions and prove theorems in all four models q C at, R ezk, S egal, 1 - C omp at once

The analytic vs synthetic theory of (∞, 1) -categories Q: How might you develop the category theory of (∞, 1) -categories? Two strategies: • work analytically to give categorical definitions and prove theorems using the combinatorics of one model (eg., Joyal, Lurie, Gepner-Haugseng, Cisinski in q C at; Kazhdan-Varshavsky, Rasekh in R ezk; Simpson in S egal) • work synthetically to give categorical definitions and prove theorems in all four models q C at, R ezk, S egal, 1 - C omp at once Our method: introduce an ∞ -cosmos to axiomatize common features of the categories q C at, R ezk, S egal, 1 - C omp of (∞, 1) -categories.

2 ∞ -cosmoi of (∞, 1) -categories

An ∞ -cosmos is a category that • is enriched over quasi-categories, i.e., functors 𝑔∶ 𝐵 → 𝐶 between ∞ -categories define the points of a quasi-category Fun (𝐵, 𝐶) , • has a class of isofibrations 𝐹 ↠ 𝐶 with familiar closure properties, • and has flexibly-weighted simplicially-enriched limits, constructed as limits of diagrams of ∞ -categories and isofibrations. Theorem. q at, ezk, egal, and 1 - omp define ∞ -cosmoi, and so do certain models of (∞, 𝑜) -categories for 0 ≤ 𝑜 ≤ ∞ , fibered versions of all of the above, and many more things besides. Henceforth ∞ -category and ∞ -functor are technical terms that mean the objects and morphisms of some ∞ -cosmos. ∞ -cosmoi of ∞ -categories Idea: An ∞ -cosmos is an “ (∞, 2) -category with (∞, 2) -categorical limits” whose objects we call ∞ -categories.

• is enriched over quasi-categories, i.e., functors 𝑔∶ 𝐵 → 𝐶 between ∞ -categories define the points of a quasi-category Fun (𝐵, 𝐶) , • has a class of isofibrations 𝐹 ↠ 𝐶 with familiar closure properties, • and has flexibly-weighted simplicially-enriched limits, constructed as limits of diagrams of ∞ -categories and isofibrations. Theorem. q at, ezk, egal, and 1 - omp define ∞ -cosmoi, and so do certain models of (∞, 𝑜) -categories for 0 ≤ 𝑜 ≤ ∞ , fibered versions of all of the above, and many more things besides. Henceforth ∞ -category and ∞ -functor are technical terms that mean the objects and morphisms of some ∞ -cosmos. ∞ -cosmoi of ∞ -categories Idea: An ∞ -cosmos is an “ (∞, 2) -category with (∞, 2) -categorical limits” whose objects we call ∞ -categories. An ∞ -cosmos is a category that

• has a class of isofibrations 𝐹 ↠ 𝐶 with familiar closure properties, • and has flexibly-weighted simplicially-enriched limits, constructed as limits of diagrams of ∞ -categories and isofibrations. Theorem. q at, ezk, egal, and 1 - omp define ∞ -cosmoi, and so do certain models of (∞, 𝑜) -categories for 0 ≤ 𝑜 ≤ ∞ , fibered versions of all of the above, and many more things besides. Henceforth ∞ -category and ∞ -functor are technical terms that mean the objects and morphisms of some ∞ -cosmos. ∞ -cosmoi of ∞ -categories Idea: An ∞ -cosmos is an “ (∞, 2) -category with (∞, 2) -categorical limits” whose objects we call ∞ -categories. An ∞ -cosmos is a category that • is enriched over quasi-categories, i.e., functors 𝑔∶ 𝐵 → 𝐶 between ∞ -categories define the points of a quasi-category Fun (𝐵, 𝐶) ,

• and has flexibly-weighted simplicially-enriched limits, constructed as limits of diagrams of ∞ -categories and isofibrations. Theorem. q at, ezk, egal, and 1 - omp define ∞ -cosmoi, and so do certain models of (∞, 𝑜) -categories for 0 ≤ 𝑜 ≤ ∞ , fibered versions of all of the above, and many more things besides. Henceforth ∞ -category and ∞ -functor are technical terms that mean the objects and morphisms of some ∞ -cosmos. ∞ -cosmoi of ∞ -categories Idea: An ∞ -cosmos is an “ (∞, 2) -category with (∞, 2) -categorical limits” whose objects we call ∞ -categories. An ∞ -cosmos is a category that • is enriched over quasi-categories, i.e., functors 𝑔∶ 𝐵 → 𝐶 between ∞ -categories define the points of a quasi-category Fun (𝐵, 𝐶) , • has a class of isofibrations 𝐹 ↠ 𝐶 with familiar closure properties,

Theorem. q at, ezk, egal, and 1 - omp define ∞ -cosmoi, and so do certain models of (∞, 𝑜) -categories for 0 ≤ 𝑜 ≤ ∞ , fibered versions of all of the above, and many more things besides. Henceforth ∞ -category and ∞ -functor are technical terms that mean the objects and morphisms of some ∞ -cosmos. ∞ -cosmoi of ∞ -categories Idea: An ∞ -cosmos is an “ (∞, 2) -category with (∞, 2) -categorical limits” whose objects we call ∞ -categories. An ∞ -cosmos is a category that • is enriched over quasi-categories, i.e., functors 𝑔∶ 𝐵 → 𝐶 between ∞ -categories define the points of a quasi-category Fun (𝐵, 𝐶) , • has a class of isofibrations 𝐹 ↠ 𝐶 with familiar closure properties, • and has flexibly-weighted simplicially-enriched limits, constructed as limits of diagrams of ∞ -categories and isofibrations.