Involutive factorisation systems & Dold-Kan correspondences - PowerPoint PPT Presentation

Involutive factorisation systems & Dold-Kan correspondences Involutive factorisation systems & Dold-Kan correspondences Clemens Berger 1 University of Nice CT 2019 Edinburgh, July 11, 2019 1 joint with Christophe Cazanave and Ingo Waschkies

Involutive factorisation systems & Dold-Kan correspondences Introduction 1 Simplicial objects 2 Involutive factorisation systems 3 Dold-Kan correspondences 4 Joyal’s categories Θ n 5

Involutive factorisation systems & Dold-Kan correspondences Introduction Theorem (Dold 1958, Kan 1958) M : Ab ∆ op ≃ Ch ( Z ) : K Corollary There is a simplicial abelian group K ( A , n ) such that π n ( K ( A , n )) = A and π i ( K ( A , n )) = 0 for i � = n . Proof. K : Ch ( Z ) → Ab ∆ op takes homology into homotopy. K ( A , n ) is n the image of the chain complex: 0 ← · · · ← 0 ← A ← 0 ← · · · Purpose of the talk Categorical structure of ∆ inducing Dold-Kan correspondence.

Involutive factorisation systems & Dold-Kan correspondences Introduction Theorem (Dold 1958, Kan 1958) M : Ab ∆ op ≃ Ch ( Z ) : K Corollary There is a simplicial abelian group K ( A , n ) such that π n ( K ( A , n )) = A and π i ( K ( A , n )) = 0 for i � = n . Proof. K : Ch ( Z ) → Ab ∆ op takes homology into homotopy. K ( A , n ) is n the image of the chain complex: 0 ← · · · ← 0 ← A ← 0 ← · · · Purpose of the talk Categorical structure of ∆ inducing Dold-Kan correspondence.

Involutive factorisation systems & Dold-Kan correspondences Introduction Theorem (Dold 1958, Kan 1958) M : Ab ∆ op ≃ Ch ( Z ) : K Corollary There is a simplicial abelian group K ( A , n ) such that π n ( K ( A , n )) = A and π i ( K ( A , n )) = 0 for i � = n . Proof. K : Ch ( Z ) → Ab ∆ op takes homology into homotopy. K ( A , n ) is n the image of the chain complex: 0 ← · · · ← 0 ← A ← 0 ← · · · Purpose of the talk Categorical structure of ∆ inducing Dold-Kan correspondence.

Involutive factorisation systems & Dold-Kan correspondences Introduction Theorem (Dold 1958, Kan 1958) M : Ab ∆ op ≃ Ch ( Z ) : K Corollary There is a simplicial abelian group K ( A , n ) such that π n ( K ( A , n )) = A and π i ( K ( A , n )) = 0 for i � = n . Proof. K : Ch ( Z ) → Ab ∆ op takes homology into homotopy. K ( A , n ) is n the image of the chain complex: 0 ← · · · ← 0 ← A ← 0 ← · · · Purpose of the talk Categorical structure of ∆ inducing Dold-Kan correspondence.

Involutive factorisation systems & Dold-Kan correspondences Introduction Theorem (Dold 1958, Kan 1958) M : Ab ∆ op ≃ Ch ( Z ) : K Corollary There is a simplicial abelian group K ( A , n ) such that π n ( K ( A , n )) = A and π i ( K ( A , n )) = 0 for i � = n . Proof. K : Ch ( Z ) → Ab ∆ op takes homology into homotopy. K ( A , n ) is n the image of the chain complex: 0 ← · · · ← 0 ← A ← 0 ← · · · Purpose of the talk Categorical structure of ∆ inducing Dold-Kan correspondence.

� � � Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (simplex category ∆) Ob ∆ = { [ n ] = { 0 , 1 . . . , n } , n ≥ 0 } , Mor ∆ = { monotone maps } Remark (epi-mono factorisation system) The category ∆ is generated by elementary face operators ǫ n i : [ n − 1] → [ n ] , 0 ≤ i ≤ n , and degeneracy operators η n i : [ n + 1] → [ n ] , 0 ≤ i ≤ n . Every simplicial operator φ : [ m ] → [ n ] factors as φ [ m ] [ n ] mono epi � � [ p ] and every epi (resp. mono)morphism in ∆ is a canonical composite of elementary degeneracy (resp. face) operators.

� � � Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (simplex category ∆) Ob ∆ = { [ n ] = { 0 , 1 . . . , n } , n ≥ 0 } , Mor ∆ = { monotone maps } Remark (epi-mono factorisation system) The category ∆ is generated by elementary face operators ǫ n i : [ n − 1] → [ n ] , 0 ≤ i ≤ n , and degeneracy operators η n i : [ n + 1] → [ n ] , 0 ≤ i ≤ n . Every simplicial operator φ : [ m ] → [ n ] factors as φ [ m ] [ n ] mono epi � � [ p ] and every epi (resp. mono)morphism in ∆ is a canonical composite of elementary degeneracy (resp. face) operators.

� � � Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (simplex category ∆) Ob ∆ = { [ n ] = { 0 , 1 . . . , n } , n ≥ 0 } , Mor ∆ = { monotone maps } Remark (epi-mono factorisation system) The category ∆ is generated by elementary face operators ǫ n i : [ n − 1] → [ n ] , 0 ≤ i ≤ n , and degeneracy operators η n i : [ n + 1] → [ n ] , 0 ≤ i ≤ n . Every simplicial operator φ : [ m ] → [ n ] factors as φ [ m ] [ n ] mono epi � � [ p ] and every epi (resp. mono)morphism in ∆ is a canonical composite of elementary degeneracy (resp. face) operators.

� � � Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (simplex category ∆) Ob ∆ = { [ n ] = { 0 , 1 . . . , n } , n ≥ 0 } , Mor ∆ = { monotone maps } Remark (epi-mono factorisation system) The category ∆ is generated by elementary face operators ǫ n i : [ n − 1] → [ n ] , 0 ≤ i ≤ n , and degeneracy operators η n i : [ n + 1] → [ n ] , 0 ≤ i ≤ n . Every simplicial operator φ : [ m ] → [ n ] factors as φ [ m ] [ n ] mono epi � � [ p ] and every epi (resp. mono)morphism in ∆ is a canonical composite of elementary degeneracy (resp. face) operators.

� � � Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (simplex category ∆) Ob ∆ = { [ n ] = { 0 , 1 . . . , n } , n ≥ 0 } , Mor ∆ = { monotone maps } Remark (epi-mono factorisation system) The category ∆ is generated by elementary face operators ǫ n i : [ n − 1] → [ n ] , 0 ≤ i ≤ n , and degeneracy operators η n i : [ n + 1] → [ n ] , 0 ≤ i ≤ n . Every simplicial operator φ : [ m ] → [ n ] factors as φ [ m ] [ n ] mono epi � � [ p ] and every epi (resp. mono)morphism in ∆ is a canonical composite of elementary degeneracy (resp. face) operators.

� � � Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (simplex category ∆) Ob ∆ = { [ n ] = { 0 , 1 . . . , n } , n ≥ 0 } , Mor ∆ = { monotone maps } Remark (epi-mono factorisation system) The category ∆ is generated by elementary face operators ǫ n i : [ n − 1] → [ n ] , 0 ≤ i ≤ n , and degeneracy operators η n i : [ n + 1] → [ n ] , 0 ≤ i ≤ n . Every simplicial operator φ : [ m ] → [ n ] factors as φ [ m ] [ n ] mono epi � � [ p ] and every epi (resp. mono)morphism in ∆ is a canonical composite of elementary degeneracy (resp. face) operators.

Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (geometric realisation, Milnor 1957) ∆ ֒ → Top : [ n ] �→ ∆ n yields by left Kan extension along Yoneda |−| ∆ : Sets ∆ op → Top . Theorem (Quillen 1968) Geometric realisation is left part of a Quillen equivalence. Definition (simplicial homology, Eilenberg 1944) N Sets ∆ op � Ab ∆ op � Ch ( Z ) � Ab N � ( N • ( X ) , d • ) ✤ � Z [ X • ] ✤ � H • ( X ) X • ✤ k ( − 1) k X ( ǫ n where ( N n ( X ) = Z [ X n ] / Z [ D n ( X )] , d n = � k )) is isomorphic to the 0 ≤ k < n X ( ǫ n k ) , d n = X ( ǫ n Moore chain complex ( M n ( X ) = � n )).

Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (geometric realisation, Milnor 1957) ∆ ֒ → Top : [ n ] �→ ∆ n yields by left Kan extension along Yoneda |−| ∆ : Sets ∆ op → Top . Theorem (Quillen 1968) Geometric realisation is left part of a Quillen equivalence. Definition (simplicial homology, Eilenberg 1944) N Sets ∆ op � Ab ∆ op � Ch ( Z ) � Ab N � ( N • ( X ) , d • ) ✤ � Z [ X • ] ✤ � H • ( X ) X • ✤ k ( − 1) k X ( ǫ n where ( N n ( X ) = Z [ X n ] / Z [ D n ( X )] , d n = � k )) is isomorphic to the 0 ≤ k < n X ( ǫ n k ) , d n = X ( ǫ n Moore chain complex ( M n ( X ) = � n )).

Involutive factorisation systems & Dold-Kan correspondences Simplicial objects Definition (geometric realisation, Milnor 1957) ∆ ֒ → Top : [ n ] �→ ∆ n yields by left Kan extension along Yoneda |−| ∆ : Sets ∆ op → Top . Theorem (Quillen 1968) Geometric realisation is left part of a Quillen equivalence. Definition (simplicial homology, Eilenberg 1944) N Sets ∆ op � Ab ∆ op � Ch ( Z ) � Ab N � ( N • ( X ) , d • ) ✤ � Z [ X • ] ✤ � H • ( X ) X • ✤ k ( − 1) k X ( ǫ n where ( N n ( X ) = Z [ X n ] / Z [ D n ( X )] , d n = � k )) is isomorphic to the 0 ≤ k < n X ( ǫ n k ) , d n = X ( ǫ n Moore chain complex ( M n ( X ) = � n )).