The Uniqueness of the Homotopy Theory of Higher Categories

Examples (n=1)

• topological categories
• simplicial categories
• topological A-infinity categories
• Segal n-categories

I think this is about monads!

Category

A Problem...

I think this is about enrichment!

???

(n=1)

The Homotopy Theory

of Homotopy Theories

Batanin

Monadic

weak algebras

for strict

n-cat monod

I think this is about presheaves!

"Trimble-May"

n-categories

Different

perspectives

M-enriched

categories

(∞,n)-Cats

Enriched

[Bergner, Lurie]

[Hirschowitz-Simpson, Pellissier]

n-fold complete

Segal Spaces

[Barwick]

Segal

n-Categories

weak

complicial

sets

[Verity]

Presheaves

complete

Segal

Θ-spaces

[Rezk]

[Joyal]

presheaves

with tranversality

on manifolds

n-quasicategories

[Ayala-Rozenblyum]

What is an (∞, n)-category?

• Complete Segal Spaces
• Quasicategories

Examples (n=1)

Homotopy Hypothesis

Informally...

an (∞, n)-category is an

∞-category with all morphism

above n invertible

The homotopy theory of groupoids

is the same as

the homotopy theory of spaces

Classically: 1-types vs 1-groupoids

What is a higher category?

an equivalence between

1-types and 1-groupoids

Informally...

The Unicity of the

Homotopy Theory of

Higher Categories

fundamental groupoid

Chris Schommer-Pries (MIT)

AMS Joint Meetings 2012

arXiv:1112.0040

Joint with

Clark Barwick

and in higher dimensions

an equivalence of homotopy theories

strong generation

Connecting to Model Categories

(∞, 0)-categories = spaces

other models of a homotopy theory

A Homotopy Theory is...?

functors out of C are controled!

more

generality

Important later!

Some posibilities:

• A model category (Quillen)
• A homotopical category (Dwyer-Kan-Smith)
• A category with any class of weak equivalences

Kan extension!

simplicially enriched category

Hammock Localization

The Proof

doesn't commute!

A canonical subcategory?

Corollary: n-fold complete Segal spaces are Quillen equivalent to Rezk's complete Theta Spaces (via the obvious map).

Consequences

Lemma: The 0-truncated objects are the gaunt n-categories

or one of the other models

Lesson:

work inside quasicategories!

Correspondences

Main Theorem

Toën's theorems

Examples

Our Axioms

Kan complex

Cells

The category Upsilon

fundamental building

blocks of n-categories

Axioms (precise version)

Hammock Localization

• generated under homotopy colimits by basic n-categories (built from cells)

• internal homs for all correspondences

• a finite list of certain obvious pushouts are actually pushouts

• universal with respect to these axioms.

may view each model category as a simplicially enriched category

leftovers

• n-trivial weak complicial sets (Street-Verity)
• dendroidally based theories (Moerdijk-Weiss)
• non-homotopical theories

commutes as a diagram

in simplicial categories

(up to higher coherent homotopy)

any other is a localization of C

The End

Thank you!

Segal

n-Categories