New model structures on simplicial sets Matt Feller University of - PowerPoint PPT Presentation

New model structures on simplicial sets Matt Feller University of Virginia CT 2019 — Edinburgh

Outline 1 Background 2 Cisinski’s Theory 3 New Stuff

Category Theory in Simplicial Sets • 0-simplices: ‚

Category Theory in Simplicial Sets • 0-simplices: ‚ 0 1 • 1-simplices:

Category Theory in Simplicial Sets • 0-simplices: ‚ 0 1 • 1-simplices: • 2-simplices: 1 0 2

Category Theory in Simplicial Sets • 0-simplices: ‚ 0 1 • 1-simplices: • 2-simplices: 1 1 Ñ ã 0 2 0 2

Category Theory in Simplicial Sets • 0-simplices: ‚ 0 1 • 1-simplices: • 2-simplices: 1 1 Ñ ã 0 2 0 2 • 3-simplices: 1 3 0 2

Category Theory in Simplicial Sets • 0-simplices: ‚ 0 1 • 1-simplices: • 2-simplices: 1 1 Ñ ã 0 2 0 2 • 3-simplices: 1 1 3 Ñ 0 3 ã 0 2 2

Category Theory in Simplicial Sets • 0-simplices: ‚ 0 1 • 1-simplices: • 2-simplices: 1 1 Ñ ã 0 2 0 2 • 3-simplices: 1 1 3 Ñ 0 3 ã 0 2 2 • (etc.)

Nerves of Categories

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows:

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C • N p C q 2 = pairs of composible morphisms in C

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C • N p C q 2 = pairs of composible morphisms in C • N p C q 3 = triples of composible morphisms in C

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C • N p C q 2 = pairs of composible morphisms in C • N p C q 3 = triples of composible morphisms in C . . .

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C • N p C q 2 = pairs of composible morphisms in C • N p C q 3 = triples of composible morphisms in C . . . Ñ Set ∆ op N : Cat ã (full/faithful)

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C • N p C q 2 = pairs of composible morphisms in C • N p C q 3 = triples of composible morphisms in C . . . Ñ Set ∆ op N : Cat ã (full/faithful) Examples g 1 g n ∆ r n s “ N p‚ ‚q . . .

Nerves of Categories Definition Given a small category C , define a simplicial set N p C q as follows: • N p C q 0 = objects of C • N p C q 1 = morphisms of C • N p C q 2 = pairs of composible morphisms in C • N p C q 3 = triples of composible morphisms in C . . . Ñ Set ∆ op N : Cat ã (full/faithful) Examples g -1 g 1 g n ∆ r n s “ N p‚ ‚q . . . J : “ N p‚ ‚q g

Categories are “Strict” The inclusion S p r n s ã Ñ ∆ r n s 1 1 Ñ 3 3 ã 0 0 2 2 is called a spine extension .

Categories are “Strict” The inclusion S p r n s ã Ñ ∆ r n s 1 1 Ñ 3 3 ã 0 0 2 2 is called a spine extension . Unique Lifting = “Strict” Categories are simplicial sets with unique spine extensions. S p r n s X X – N p C q ð ñ D ! (for some C ) ∆ r n s n ě 2

Categories are “1-Segal Sets” Unique lifting condition for categories is 1-dimensional: S p r n s is composed of 1-simplices.

Categories are “1-Segal Sets” Unique lifting condition for categories is 1-dimensional: S p r n s is composed of 1-simplices. Interpretation Categories are “1-Segal sets.”

Categories are “1-Segal Sets” Unique lifting condition for categories is 1-dimensional: S p r n s is composed of 1-simplices. Interpretation Categories are “1-Segal sets.” What are 2-Segal sets? • More general than categories

Categories are “1-Segal Sets” Unique lifting condition for categories is 1-dimensional: S p r n s is composed of 1-simplices. Interpretation Categories are “1-Segal sets.” What are 2-Segal sets? • More general than categories • Unique “2-dimensional spine extensions”

Categories are “1-Segal Sets” Unique lifting condition for categories is 1-dimensional: S p r n s is composed of 1-simplices. Interpretation Categories are “1-Segal sets.” What are 2-Segal sets? • More general than categories • Unique “2-dimensional spine extensions” still “strict” è

2-Segal Sets Triangulations of the square: T 1 : T : 3 2 3 2 0 1 0 1

2-Segal Sets Triangulations of the square: T 1 : T : 3 2 3 2 0 1 0 1 1 1 3 3 0 0 2 2

2-Segal Sets Triangulations of the hexagon: 4 3 4 3 T 1 : T : 5 2 5 2 0 1 0 1 (etc.)

2-Segal Sets Intuition Think of the inclusions T ã Ñ ∆ r n s as “2-dimensional spine extensions.”

2-Segal Sets Intuition Think of the inclusions T ã Ñ ∆ r n s as “2-dimensional spine extensions.” Definition A 2-Segal set is a simplicial set X with a unique lifting condition: T X ð ñ X is 2-Segal D ! ∆ r n s n ě 3

2-Segal Sets Examples • (Nerves of) categories are 2-Segal.

2-Segal Sets Examples • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.)

2-Segal Sets Examples • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.) • Output of Waldhausen S ‚ construction (from algebraic K-theory) applied to nice enough double categories.

2-Segal Sets Examples • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.) • Output of Waldhausen S ‚ construction (from algebraic K-theory) applied to nice enough double categories. • Lots of other examples from combinatorics.

2-Segal Sets Examples • (Nerves of) categories are 2-Segal. (1-Segal ñ 2-Segal.) • Output of Waldhausen S ‚ construction (from algebraic K-theory) applied to nice enough double categories. • Lots of other examples from combinatorics. Another Perspective 2-Segal sets are equivalent to multivalued categories , where composition is not always unique or defined, but is associative.

Homotopical/ 8 /Non-Strict Versions Categories have a homotopical analogue in simplicial sets.

Homotopical/ 8 /Non-Strict Versions Categories have a homotopical analogue in simplicial sets. Definition A quasi-category is a simplicial set X with a (non-unique) lifting condition: Λ i r n s X ð ñ X is a D quasi-category ∆ r n s 0 ă i ă n

Homotopical/ 8 /Non-Strict Versions Categories have a homotopical analogue in simplicial sets. Definition A quasi-category is a simplicial set X with a (non-unique) lifting condition: Λ i r n s X ð ñ X is a D quasi-category ∆ r n s 0 ă i ă n • Composition is always defined, but only “unique up to homotopy.”

Homotopical/ 8 /Non-Strict Versions Categories have a homotopical analogue in simplicial sets. Definition A quasi-category is a simplicial set X with a (non-unique) lifting condition: Λ i r n s X ð ñ X is a D quasi-category ∆ r n s 0 ă i ă n • Composition is always defined, but only “unique up to homotopy.” • Special Case: If all morphisms are invertible, we have a Kan complex —also defined by a non-unique lifting condition.

Question Strict Homotopical Category

Question Strict Homotopical Category Quasi-category

Question Strict Homotopical Groupoid Category Quasi-category

Question Strict Homotopical Groupoid Kan Complex Category Quasi-category

Question Strict Homotopical Groupoid Kan Complex Category Quasi-category 2-Segal Set

Question Strict Homotopical Groupoid Kan Complex Category Quasi-category 2-Segal Set ???

Model Structure = “Homotopy Theory” We can endow a category with a “homotopy theory” by putting a model structure on it.

Model Structure = “Homotopy Theory” We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects , defined by a lifting condition.

Model Structure = “Homotopy Theory” We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects , defined by a lifting condition. Examples • Classical model structure on Set ∆ op :

Model Structure = “Homotopy Theory” We can endow a category with a “homotopy theory” by putting a model structure on it. Every model structure comes with a class of well-behaved objects, called the fibrant objects , defined by a lifting condition. Examples • Classical model structure on Set ∆ op : equivalent to homotopy theory of topological spaces è