The homotopy relation in a category with weak equivalences Martin - PowerPoint PPT Presentation

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories The homotopy relation in a category with weak equivalences Martin Szyld University of Buenos Aires - CONICET, Argentina CT 2018 @ UA¸ c, Ponta Delgada, Portugal

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Model (bi)categories: a structure ( C , F , co F , W ), with C a (bi)category, and F co F W families of arrows of C · � � ∼ � � · � · � · · · satisfying some axioms.

� � � �� � � �� � � �� � � �� � �� � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Model (bi)categories: a structure ( C , F , co F , W ), with C a (bi)category, and F co F W families of arrows of C · � � ∼ � � · � · � · · · satisfying some axioms. A taste of the axioms: � · · · · · · � � � � ∼ ∼ and � · · · · · · · · � � � � � � or ∼ ∼ � � · � · � · · · ·

� � � �� � � �� � � �� � � �� � �� � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Model (bi)categories: a structure ( C , F , co F , W ), with C a (bi)category, and F co F W families of arrows of C · � � ∼ � � · � · � · · · satisfying some axioms. A taste of the axioms: � · · · · · · � � � � ∼ ∼ ∼ ∼ and = = � · · · · · · · · � � � � � � or ∼ ∼ � � · � · � · · · ·

� � �� �� � � � � �� �� � � �� � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Model (bi)categories: a structure ( C , F , co F , W ), with C a (bi)category, and F co F W families of arrows of C · � � ∼ � � · � · � · · · satisfying some axioms. A taste of the axioms: � · · · · · · � � � � ∼ ∼ ∼ ∼ and = = � · · · · · · · · � � � � � � ∼ or ∼ = = ∼ ∼ � � · � · � · · · ·

�� �� � � � � �� � �� � �� � � � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Model (bi)categories: a structure ( C , F , co F , W ), with C a (bi)category, and F co F W families of arrows of C · � � ∼ � � · � · � · · · satisfying some axioms. A taste of the axioms: � · · · · · · � � � � ∼ ∼ ∼ ∼ and = = � · · · · · · · · � � � � � � ∼ = ∼ or ∼ = = ∼ ∼ � ∼ = � · � · � · · · ·

�� �� � � � � �� � �� � �� � � � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Model (bi)categories: a structure ( C , F , co F , W ), with C a (bi)category, and F co F W families of arrows of C · � � ∼ � � · � · � · · · satisfying some axioms. A taste of the axioms: � · · · · · · � � � � ∼ ∼ ∼ ∼ and = = � · · · · · · · · � � � � � � ∼ = ∼ or ∼ = = ∼ ∼ � ∼ = � · � · � · · · · Ho( C ) = C [ W − 1 ] admits a construction “quotienting by homotopy”.

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Our original problem: homotopy in a model bicategory We 1 seek a construction of the homotopy bicategory H o( C ): - Objects and arrows are those of C fc ( 0 � � � X � � 1 ). - 2-cells: classes [ H ] of “homotopies” by an eq. relation. 1 together with E. Descotte and E. Dubuc.

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Our original problem: homotopy in a model bicategory We 1 seek a construction of the homotopy bicategory H o( C ): - Objects and arrows are those of C fc ( 0 � � � X � � 1 ). - 2-cells: classes [ H ] of “homotopies” by an eq. relation. Simultaneous requirements � - Vertical composition compatible with the eq. relation - Horizontal composition - (Non invertible) 2-cell �→ homotopy 1 together with E. Descotte and E. Dubuc.

� � � � � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Our original problem: homotopy in a model bicategory We 1 seek a construction of the homotopy bicategory H o( C ): - Objects and arrows are those of C fc ( 0 � � � X � � 1 ). - 2-cells: classes [ H ] of “homotopies” by an eq. relation. Simultaneous requirements � - Vertical composition compatible with the eq. relation - Horizontal composition - (Non invertible) 2-cell �→ homotopy Considering Quillen’s notion � an obstacle f ℓ A B f ∼ g if and only if there is a diagram g ∂ 0 in which σ is a weak equivalence (and id h ∂ 1 ∂ 0 + ∂ 1 A ∐ A − − − − → A × I is a cofibration) A ∼ A × I σ 1 together with E. Descotte and E. Dubuc.

� � � � � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Our original problem: homotopy in a model bicategory We 1 seek a construction of the homotopy bicategory H o( C ): - Objects and arrows are those of C fc ( 0 � � � X � � 1 ). - 2-cells: classes [ H ] of “homotopies” by an eq. relation. Simultaneous requirements � - Vertical composition compatible with the eq. relation - Horizontal composition - (Non invertible) 2-cell �→ homotopy Considering Quillen’s notion � an obstacle f ℓ ℓ A B f ∼ g ⇒ jf ∼ jg ✓ g ∂ 0 id h ∂ 1 A ∼ A × I σ 1 together with E. Descotte and E. Dubuc.

� � � � � � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Our original problem: homotopy in a model bicategory We 1 seek a construction of the homotopy bicategory H o( C ): - Objects and arrows are those of C fc ( 0 � � � X � � 1 ). - 2-cells: classes [ H ] of “homotopies” by an eq. relation. Simultaneous requirements � - Vertical composition compatible with the eq. relation - Horizontal composition - (Non invertible) 2-cell �→ homotopy Considering Quillen’s notion � an obstacle f j ℓ ℓ A ′ A B f ∼ g ⇒ jf ∼ jg ✓ g ∂ 0 ℓ ℓ f ∼ g ⇒ fj ∼ gj : id h ∂ 1 A ∼ A × I σ 1 together with E. Descotte and E. Dubuc.

� � � � � � � Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Our original problem: homotopy in a model bicategory We 1 seek a construction of the homotopy bicategory H o( C ): - Objects and arrows are those of C fc ( 0 � � � X � � 1 ). - 2-cells: classes [ H ] of “homotopies” by an eq. relation. Simultaneous requirements � - Vertical composition compatible with the eq. relation - Horizontal composition - (Non invertible) 2-cell �→ homotopy Considering Quillen’s notion � an obstacle f j ℓ ℓ A ′ A B f ∼ g ⇒ jf ∼ jg ✓ g ∂ 0 ℓ ℓ f ∼ g ⇒ fj ∼ gj : h � id ∂ 1 ℓ r r ℓ f ∼ g ⇒ f ∼ g ⇒ fj ∼ gj ⇒ fj ∼ gj B I ∼ B σ 1 together with E. Descotte and E. Dubuc.

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Homotopy in a category with weak equivalences Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious.

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Homotopy in a category with weak equivalences Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. � Section 1: model categories, Section 2: categories with weak equivalences ( C , W ). ℓ r Section 1: Ho( C fc ) = C fc / ∼ , with ∼ = ∼ = ∼ “long and technical”

Motivation The relation ∼W and Whitehead Constructing ∼W The case of model categories Homotopy in a category with weak equivalences Quote from [DHKS] book Many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. � Section 1: model categories, Section 2: categories with weak equivalences ( C , W ). ℓ r Section 1: Ho( C fc ) = C fc / ∼ , with ∼ = ∼ = ∼ “long and technical” Considering ∼ W for ( C , W ) simplifies and clarifies this argument