Semistrict models of connected 3-types and Tamsamanis weak - PDF document

Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids Simona Paoli, Macquarie University

Main themes • Modelling connected 3-types: cat 2 -groups (Loday). Homotopy theory ✲ ✲ Tamsamani’s weak Higher category theory 3-groupoids (with 1 object) • Comparison problem. • Semistrictification results for Tamsamani’s weak 3-groupoids with 1 object.

Cat n -groups as homotopy models Cat n (Gp) = Cat (Cat n − 1 (Gp)) • Definition Cat 0 (Gp) = Gp N : Cat n (Gp) → [∆ n op , Gp] • Multinerve • Classifying space of G ∈ Cat n (Gp) B G = B NG . f : G → G ′ mor(Cat n (Gp)) • Weak equivalence s.t. Bf weak homotopy equivalence. • Theorem [Whitehead n = 1] [Loday; Bullejos-Cegarra-Duskin; Porter, n ≥ 1] � connected B : Cat n (Gp) � ≃ H o : P n + 1-types ∼

Tamsamani’s model: n=2 • Segal maps C category with finite limits, φ ∈ [∆ op , C ] η n : φ n → φ 1 × φ 0 · · · n × φ 0 φ 1 . n ≥ 2 fact: φ nerve of object of Cat C ⇔ η n isomorphism for all n ≥ 2. • Tamsamani’s weak 2-nerves, N 2 . φ ∈ [∆ 2 op , Set] φ n = ([ n ] , -) (i) φ n nerve of category of all n ≥ 0. (ii) φ 0 constant. (iii) Segal maps equivalences of categories ∀ n ≥ 2. • Weak 2-groupoids T 2 , φ ∈ N 2 s.t. (i) φ n nerve of groupoid, ∀ n ≥ 0. (ii) Tφ : ∆ op → Set nerve of groupoid ( Tφ ) n = π 0 φ n • External equivalences of 2-nerves f : φ → ψ φ 1 = � φ ( x,y ) x,y ∈ φ 0 (i) φ ( x,y ) → ψ ( fx,fy ) (ii) Tf equivalences of categories.

Tamsamani’s model: n=3 • Tamsamani’s weak 3-nerves, N 3 . φ ∈ [∆ 3 op , Set] φ n = ([ n ] , - , -) (i) φ n ∈ N 2 ∀ n ≥ 0. (ii) φ 0 constant. (iii) Segal maps equivalences of 2-nerves ∀ n ≥ 2. • Weak 3-groupoids T 3 , φ ∈ N 3 s.t. (i) φ n ∈ T 2 ∀ n ≥ 0. (ii) T 2 φ : ∆ op → Set nerve of groupoid. • Fact: external equivalences in T 2 and T 3 ≡ weak homotopy equivalences • The subcategory S ⊂ T 3 φ ∈ S if φ ∈ T 3 and φ 0 (- , -) = {·} . • Theorem [Tamsamani] T 3 / ∼ ext ≃ H o (3-types) � � connected S / ∼ ext ≃ H o 3-types

Summary: cat 2 -gp versus T 3 . Cat 2 (Gp) T 3 • G ∈ [∆ 2 op , Gp] • φ ∈ [∆ 3 op , Set] G n nerve of Cat (Gp) φ n ∈ T 2 Segal maps iso. φ 0 constant, Tφ iso. Segal maps equiva- lences • multisimplicial • multisimplicial inductive definition inductive definition based on Gp based on Set strict structure weak structure “cubical” “globular” • Main issues in the comparison: discretization ✲ globular cubical nerve ✲ [∆ op , Set] Gp • dealt with functors: disc Cat 2 (Gp) / ∼ ✲ D / ∼ ✲ H / ∼ ext D / ∼ H ⊂ S .

The discretization functor • Key Lemma: G ∈ Cat 2 (Gp). There is φ ∈ Cat 2 (Gp) ∂ 0 ✲ c ✲ φ 1 φ 1 × φ 0 φ 1 ∂ 1 ✲ φ 0 ✛ σ 0 with φ 0 projective in Cat (Gp) and Bφ ≃ B G . • Projective objects in Cat (Gp) ✲ φ d d : φ 0 0 weak equivalence. φ d 0 discrete internal category. ✲ φ 0 , section t : φ d dt = id. 0 • The discrete multinerve ds N φ ∈ [∆ 2 op , Gp] d∂ 0 ✲ ✲ φ d · · · φ 1 × φ 0 φ 1 ✲ φ 1 d∂ 1 ✲ ✛ 0 ✲ ✛ ✛ σ 0 t i) B ds N φ = Bφ ≃ B G . ii) Segal maps weak equivalences in [∆ op , Gp]. ✲ D / ∼ disc : Cat 2 (Gp) / ∼ • Functor disc [ G ] = [ ds N φ ] D ⊂ [∆ 2 op , Gp] “internal 2-nerves”.

� � First semistrictification result. • The subcategory H ⊂ S . φ ∈ S and Segal maps φ n → φ 1 × · · · n × φ 1 iso. Objects of H are “semistrict”. • Theorem [P.] Commutative diagram F Cat 2 (Gp) / ∼ H / ∼ ext � � � � � � � � � � � � � � � � � � � � � � � � � B B � � � � � � � � � � � � � � � � � � connected H o 3-types disc R ✲ D / ∼ ✲ H / ∼ ext . where F : Cat 2 (Gp) / ∼ Let H o S ( H ) ⊂ S / ∼ ext full subcategory with ob- jects in H . Then Cat 2 (Gp) ≃ H o S ( H ) . ∼ • Corollary: Every object of S is equivalent to an object of H through a zig-zag of external equiva- lences. • Remark: H ⊂ Mon( T 2 , × ) .

� � Second semistrictification result. • The subcategory K ⊂ S . φ ∈ S and φ n strict 2-groupoid ∀ n ≥ 0. Objects of K are semistrict but K � = H . • Theorem[P.] Commutative diagram St S / ∼ ext K / ∼ ext � � � � � � � � � � � � � � � � � � � � � B B � � � � � � � � � � � � connected H o 3-types Let H o S ( K ) ⊂ S / ∼ ext full subcategory with ob- jects in K . Then S / ∼ ext ≃ H o S ( K ) idea of proof: G st ν ✲ T st ✲ Bigpd ✲ 2- gpd St : T 2 2 ψ ∈ S , ( St ψ ) n = St ψ n . ( St ψ ) n = St ψ n ≃ St ( ψ 1 × n · · · × ψ 1 ) ≃ ≃ St ψ 1 × · · · × St ψ 1 = ( St ψ ) 1 × · · · × ( St ψ ) 1 hence St ψ ∈ K .

� � � � The comparison with Gray groupoids. • Gray groupoids. Gray =(2-cat, ⊗ gray ). Gray-enriched category with invertible cells. • Theorem [Joyal - Tierney, Leroy] H o (3 − types) ≃ Gray - gpd/ ∼ H o (conn. 3-types) ≃ ( Gray - gpd ) 0 / ∼ . • Theorem [P.] Commutative diagram S T H o S ( H ) ( Gray - gpd ) 0 / ∼ H o S ( K ) � � � �������������������� � � � � � � � � � � B � � � � B B � � � � � � connected H o 3-types idea of proof: - Monoidal functor G st ✲ ( Bigpd, × ) ✲ (2- gpd, ⊗ gray ) ( T 2 , × ) φ ∈ H ⊂ Mon ( T 2 , × ) ⇒ st G φ ∈ ( Gray - gpd ) 0 Let S ( φ ) = st Bic φ . - Every object of K is equivalent to one of St H . T [ ψ ] = T [ St φ ] = [ st G φ ].

� � � � � � � � � Conclusion: modelling connected 3-types using Tamsamani’s model. • Tamsamani’s weak 3-groupoids, S . . . . � � . � . � � � . � � . � . � � . � . � � � . . � � � . � � � . � � . � . � � � . . . � � � ✲ . . . . . · · · . ✛ ✲ . . . . . . . ✲ ✛ . . . . . � � . � ✛ ✲ � � . � . � � � � � . � � ✲ � � . � � � � � � � � � � � � � � � strict � � weak S / ∼ ext ≃ H o connected � 3-types � � � � � � � � weak � � • Semistrict cases. a) H ⊂ S strict � � connected H o S ( H ) ≃ H o weak 3-types � � � � H ⊂ Mon ( T 2 , × ) . � � � � � strict � � b) K ⊂ S strict � � strict connected H o S ( K ) ≃ H o � 3-types � � � � � � � � weak � �