Another approach to the Kan-Quillen model structure

Another approach to the Kan-Quillen model structure

CT 2015, Aveiro

Sean Moss

Tuesday 16 June 2015

University of Cambridge, UK

Motivation

∙ Try to understand the basic homotopy theory of simplicial sets combinatorially — without topological spaces or minimal fibrations.

∙ See how much of simplicial homotopy theory still holds constructively.

∙ This talk: an elementary proof that the embedding X,→Ex^∞Xis an anodyne extension.

Motivation

∙ Try to understand the basic homotopy theory of simplicial sets combinatorially — without topological spaces or minimal fibrations.

∙ See how much of simplicial homotopy theory still holds constructively.

∙ This talk: an elementary proof that the embedding X,→Ex^∞Xis an anodyne extension.

1

Motivation

∙ Try to understand the basic homotopy theory of simplicial sets combinatorially — without topological spaces or minimal fibrations.

∙ See how much of simplicial homotopy theory still holds constructively.

∙ This talk: an elementary proof that the embedding X,→Ex^∞Xis an anodyne extension.

Motivation

∙ Try to understand the basic homotopy theory of simplicial sets combinatorially — without topological spaces or minimal fibrations.

∙ See how much of simplicial homotopy theory still holds constructively.

∙ This talk: an elementary proof that the embedding X,→Ex^∞Xis an anodyne extension.

1

Horn inclusions

∙ Horn inclusionsΛⁿ_k ,→ △ⁿ represent the basic homotopical deformations of simplicial sets.

0 1 0 1

0 2

1

0 2

1

0 2

1

∙ A mapf:X→Ywith theright lifting propertywith respect to all horn inclusions is aKan fibration.

Anodyne Extensions

∙ Ananodyne presentationfor a monomorphismm:A,→B consists of:

∙ a cardinalκ,

∙ a nested decompositionB=∪

α≤κA^αwithA^κ=B,A⁰=Aand for any non-zero limit ordinalλ≤κwe haveA^λ=∪

α<λA^α, and

∙ for eachα < κa pushout square of the form

⊔

i∈I_αΛⁿ_k^i,α

i,α

⊔

i∈Iα△^ni,α

A^α

A^α+1

∙ Astrong anodyne extensionis anmadmitting an anodyne presentation.

3

Anodyne Extensions

∙ Ananodyne presentationfor a monomorphismm:A,→B consists of:

∙ a cardinalκ,

∙ a nested decompositionB=∪

α≤κA^αwithA^κ=B,A⁰=Aand for any non-zero limit ordinalλ≤κwe haveA^λ=∪

α<λA^α, and

∙ for eachα < κa pushout square of the form

⊔

i∈I_αΛⁿ_k^i,α

i,α

⊔

i∈Iα△^ni,α

A^α

A^α+1

∙ Astrong anodyne extensionis anmadmitting an anodyne presentation.

Anodyne Extensions

∙ Ananodyne presentationfor a monomorphismm:A,→B consists of:

∙ a cardinalκ,

∙ a nested decompositionB=∪

α≤κA^αwithA^κ=B,A⁰=Aand for any non-zero limit ordinalλ≤κwe haveA^λ=∪

α<λA^α, and

∙ for eachα < κa pushout square of the form

⊔

i∈I_αΛⁿ_k^i,α

i,α

⊔

i∈Iα△^ni,α

A^α

A^α+1

∙ Astrong anodyne extensionis anmadmitting an anodyne presentation.

3

Anodyne Extensions

∙ Ananodyne presentationfor a monomorphismm:A,→B consists of:

∙ a cardinalκ,

∙ a nested decompositionB=∪

α≤κA^αwithA^κ=B,A⁰=Aand for any non-zero limit ordinalλ≤κwe haveA^λ=∪

α<λA^α, and

∙ for eachα < κa pushout square of the form

⊔

i∈I_αΛⁿ_k^i,α

i,α

⊔

i∈I_α△^ni,α

A^α

A^α+1

∙ Astrong anodyne extensionis anmadmitting an anodyne presentation.

Anodyne Extensions

∙ Ananodyne presentationfor a monomorphismm:A,→B consists of:

∙ a cardinalκ,

∙ a nested decompositionB=∪

α≤κA^αwithA^κ=B,A⁰=Aand for any non-zero limit ordinalλ≤κwe haveA^λ=∪

α<λA^α, and

∙ for eachα < κa pushout square of the form

⊔

i∈I_αΛⁿ_k^i,α

i,α

⊔

i∈I_α△^ni,α

A^α

A^α+1

∙ Astrong anodyne extensionis anmadmitting an anodyne presentation.

3

P-structures I

∙ Letm:A,→Bbe a strong anodyne extension, andτ ∈Ba non-degenerate simplex not inA(of dimension 1, say). There are two possibilities:

τ

A^α

A^α+1

τ

∙ Eitherτ is the ‘homotopy’/‘horn-filler’ (type I) or the

‘composite’/‘added face’ (type II).

P-structures I

∙ Letm:A,→Bbe a strong anodyne extension, andτ ∈Ba non-degenerate simplex not inA(of dimension 1, say). There are two possibilities:

τ

A^α

A^α+1

τ

∙ Eitherτ is the ‘homotopy’/‘horn-filler’ (type I) or the

‘composite’/‘added face’ (type II).

4

P-structures I

∙ Letm:A,→Bbe a strong anodyne extension, andτ ∈Ba non-degenerate simplex not inA(of dimension 1, say). There are two possibilities:

τ

A^α

A^α+1

τ

∙ Eitherτ is the ‘homotopy’/‘horn-filler’ (type I) or the

‘composite’/‘added face’ (type II).

P-structures I

∙ Letm:A,→Bbe a strong anodyne extension, andτ ∈Ba non-degenerate simplex not inA(of dimension 1, say). There are two possibilities:

τ

A^α

A^α+1

τ

∙ Eitherτ is the ‘homotopy’/‘horn-filler’ (type I) or the

‘composite’/‘added face’ (type II).

4

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover),x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne presentation.

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover),x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne presentation.

5

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover),x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne presentation.

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover),x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne presentation.

5

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover), x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne presentation.

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover), x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne presentation.

5

P-structures II

Definition

AP-structurefor a monomorphismm:A,→Bconsists of:

∙ a partitionB_n.d.\A_n.d.=B_I⊔B_IIand

∙ a bijectionP:B_II→B_I such that

∙ dimP(x) =dimx+1,

∙ (moreover), x=d_iP(x)for a uniquei, and

∙ “each x∈B_n.d.has only finitely many ancestors”.

Theorem

m admits a P-structure if and only if m admits an anodyne

Subdivision

∙ Definesubdivisionsd:△ →sSet by sd(△ⁿ) =N(P_̸=∅([n])) (NBP_̸=∅ ∼= (△ⁿ)_n.d.).

0 0 01 1 0 1

2

01

02 12

012

∙ Define thelast-vertex mapȷ̄_n: sd△ⁿ→ △ⁿas the nerve of P_̸=∅([n])→[n]

S7→maxS.

6

Subdivision

∙ Definesubdivisionsd:△ →sSet by sd(△ⁿ) =N(P_̸=∅([n])) (NBP_̸=∅ ∼= (△ⁿ)_n.d.).

0 0 01 1 0 1

2

01

02 12

012

∙ Define thelast-vertex mapȷ̄_n: sd△ⁿ→ △ⁿas the nerve of P_̸=∅([n])→[n]

S7→maxS.

Subdivision

∙ Definesubdivisionsd:△ →sSet by sd(△ⁿ) =N(P_̸=∅([n])) (NBP_̸=∅ ∼= (△ⁿ)_n.d.).

0 0 01 1 0 1

2

01

02 12

012

∙ Define thelast-vertex mapȷ̄_n: sd△ⁿ→ △ⁿas the nerve of P_̸=∅([n])→[n]

S7→maxS.

6

Extension

∙ DefineextensionEx:sSet→sSet by(ExX)_n=sSet(sd△ⁿ,X)

— the set of all ‘subdivision pasting diagrams inX’.

∙ There is a canonical embeddingj_X:X→ExXgiven by X_n→(ExX)_n

(σ:△ⁿ→X)7→(σ◦ȷ̄_n: sd△ⁿ→X).

∙ ‘Iterating’ gives a Kan complex Ex^∞Xinto whichXembeds. Theorem

j_X:X→ExX admits an anodyne presentation. We will give a P-structure forj_X:X→ExX.

Extension

∙ DefineextensionEx:sSet→sSet by(ExX)_n=sSet(sd△ⁿ,X)

— the set of all ‘subdivision pasting diagrams inX’.

∙ There is a canonical embeddingj_X:X→ExXgiven by X_n→(ExX)_n

(σ:△ⁿ→X)7→(σ◦ȷ̄_n: sd△ⁿ→X).

∙ ‘Iterating’ gives a Kan complex Ex^∞Xinto whichXembeds. Theorem

j_X:X→ExX admits an anodyne presentation. We will give a P-structure forj_X:X→ExX.

7

Extension

∙ DefineextensionEx:sSet→sSet by(ExX)_n=sSet(sd△ⁿ,X)

— the set of all ‘subdivision pasting diagrams inX’.

∙ There is a canonical embeddingj_X:X→ExXgiven by X_n→(ExX)_n

(σ:△ⁿ→X)7→(σ◦ȷ̄_n: sd△ⁿ→X).

∙ ‘Iterating’ gives a Kan complex Ex^∞Xinto whichXembeds.

Theorem

j_X:X→ExX admits an anodyne presentation. We will give a P-structure forj_X:X→ExX.

Extension

∙ DefineextensionEx:sSet→sSet by(ExX)_n=sSet(sd△ⁿ,X)

— the set of all ‘subdivision pasting diagrams inX’.

∙ There is a canonical embeddingj_X:X→ExXgiven by X_n→(ExX)_n

(σ:△ⁿ→X)7→(σ◦ȷ̄_n: sd△ⁿ→X).

∙ ‘Iterating’ gives a Kan complex Ex^∞Xinto whichXembeds.

Theorem

j_X:X→ExX admits an anodyne presentation.

We will give a P-structure forj_X:X→ExX.

7

The complexity of a binary pasting diagram

Definition

For natural numbersnandkwith 0≤k≤n, let

j^k_n: sd△ⁿ→sd△ⁿbe the nerve of the binary-join-preserving mapP_̸=∅([n])→P_̸=∅([n])with

{i} 7→





{i} ifi≤k {0,1, . . . ,i} ifi≥k+1.

Observe thatjⁿ_n =id andj⁰_n isȷ̄_n followed by an inclusion. Sincej^k_n◦j^h_n=j^k_nforh≥k, we get a ‘filtration’ of(ExX)_n

X_n=J⁰_n ⊆J¹_n⊆. . .⊆Jⁿ_n⁻¹⊆Jⁿ_n= (ExX)_n, whereJ^k_n ={σ ∈(ExX)_n|σ◦j^k_n=σ}.

The complexity of a binary pasting diagram

Definition

For natural numbersnandkwith 0≤k≤n, let

j^k_n: sd△ⁿ→sd△ⁿbe the nerve of the binary-join-preserving mapP_̸=∅([n])→P_̸=∅([n])with

{i} 7→





{i} ifi≤k {0,1, . . . ,i} ifi≥k+1.

Observe thatjⁿ_n =id andj⁰_nisȷ̄_n followed by an inclusion.

Sincej^k_n◦j^h_n=j^k_nforh≥k, we get a ‘filtration’ of(ExX)_n X_n=J⁰_n ⊆J¹_n⊆. . .⊆Jⁿ_n⁻¹⊆Jⁿ_n= (ExX)_n,

whereJ^k_n ={σ ∈(ExX)_n|σ◦j^k_n=σ}.

8

The complexity of a binary pasting diagram

Definition

For natural numbersnandkwith 0≤k≤n, let

j^k_n: sd△ⁿ→sd△ⁿbe the nerve of the binary-join-preserving mapP_̸=∅([n])→P_̸=∅([n])with

{i} 7→





{i} ifi≤k {0,1, . . . ,i} ifi≥k+1.

Observe thatjⁿ_n =id andj⁰_nisȷ̄_n followed by an inclusion.

Sincej^k_n◦j^h_n=j^k_nforh≥k, we get a ‘filtration’ of(ExX)_n

−

Subdivision-pasting in terms of horn-filling I

Definition

For natural numbersnandkwith 1≤k≤n, let

r^k_n: sd△ⁿ⁺¹→ △ⁿbe the nerve of the binary-join-preserving mapP_̸=∅([n+1])→P_̸=∅([n])with

{i} 7→









{i} ifi≤k {0,1, . . .k} ifi=k+1 {i−1} ifi≥k+2.

Observe that this pattern continued tok=0 would give us sd(s⁰:△ⁿ⁺¹→ △ⁿ), the 0^th subdivided codegeneracy map.

9

Subdivision-pasting in terms of horn-filling II

0

1 3 2

0

1 2

0

1 2 3

0

1 2

Subdivision-pasting in terms of horn-filling II

0

1 3 2

0

1 2

r¹₂

0

1 2 3

0

1 2

r²₂

11

Subdivision-pasting in terms of horn-filling III

r¹₃ r²₃ r³₃

The P-structure

∙ Fix a simplicial setX.

∙ Letσ be a non-degeneraten-simplex of ExX, which moreover is inJ^k_n\J^k_n⁻¹ ⊆(ExX)_nfor somek≥1.

∙ Ifσis of type II, then we defineP(σ) =σ◦r^k_n, where

∙ we say that that such aσis oftype IIiff it isnotof the form τ◦r^h_nfor anyh≥1 andτ ∈J^h_n\J^h_n⁻¹ (andtype Iif it is).

∙ One now checks that this does indeed satisfy the conditions for a P-structure onj_X:X,→ExX.

∙ This description is useful for proving properties of Ex^∞X, whence there is an elementary deduction of the Kan-Quillen model structure.

13

The P-structure

∙ Fix a simplicial setX.

∙ Letσ be a non-degeneraten-simplex of ExX, which moreover is inJ^k_n\J^k_n⁻¹ ⊆(ExX)_n for somek≥1.

∙ Ifσis of type II, then we defineP(σ) =σ◦r^k_n, where

∙ we say that that such aσis oftype IIiff it isnotof the form τ◦r^h_nfor anyh≥1 andτ ∈J^h_n\J^h_n⁻¹ (andtype Iif it is).

∙ One now checks that this does indeed satisfy the conditions for a P-structure onj_X:X,→ExX.

∙ This description is useful for proving properties of Ex^∞X, whence there is an elementary deduction of the Kan-Quillen model structure.

The P-structure

∙ Fix a simplicial setX.

∙ Letσ be a non-degeneraten-simplex of ExX, which moreover is inJ^k_n\J^k_n⁻¹ ⊆(ExX)_n for somek≥1.

∙ Ifσis of type II, then we defineP(σ) =σ◦r^k_n, where

∙ we say that that such aσis oftype IIiff it isnotof the form τ◦r^h_nfor anyh≥1 andτ ∈J^h_n\J^h_n⁻¹ (andtype Iif it is).

∙ One now checks that this does indeed satisfy the conditions for a P-structure onj_X:X,→ExX.

∙ This description is useful for proving properties of Ex^∞X, whence there is an elementary deduction of the Kan-Quillen model structure.

13

The P-structure

∙ Fix a simplicial setX.

∙ Letσ be a non-degeneraten-simplex of ExX, which moreover is inJ^k_n\J^k_n⁻¹ ⊆(ExX)_n for somek≥1.

∙ Ifσis of type II, then we defineP(σ) =σ◦r^k_n, where

∙ we say that that such aσis oftype IIiff it isnotof the form τ ◦r^h_nfor anyh≥1 andτ ∈J^h_n\J^h_n⁻¹(andtype Iif it is).

∙ One now checks that this does indeed satisfy the conditions for a P-structure onj_X:X,→ExX.

∙ This description is useful for proving properties of Ex^∞X, whence there is an elementary deduction of the Kan-Quillen model structure.

The P-structure

∙ Fix a simplicial setX.

∙ Letσ be a non-degeneraten-simplex of ExX, which moreover is inJ^k_n\J^k_n⁻¹ ⊆(ExX)_n for somek≥1.

∙ Ifσis of type II, then we defineP(σ) =σ◦r^k_n, where

∙ we say that that such aσis oftype IIiff it isnotof the form τ ◦r^h_nfor anyh≥1 andτ ∈J^h_n\J^h_n⁻¹(andtype Iif it is).

∙ One now checks that this does indeed satisfy the conditions for a P-structure onj_X:X,→ExX.

∙ This description is useful for proving properties of Ex^∞X, whence there is an elementary deduction of the Kan-Quillen model structure.

13

The P-structure

∙ Fix a simplicial setX.

∙ Letσ be a non-degeneraten-simplex of ExX, which moreover is inJ^k_n\J^k_n⁻¹ ⊆(ExX)_n for somek≥1.

∙ Ifσis of type II, then we defineP(σ) =σ◦r^k_n, where

∙ we say that that such aσis oftype IIiff it isnotof the form τ ◦r^h_nfor anyh≥1 andτ ∈J^h_n\J^h_n⁻¹(andtype Iif it is).

∙ One now checks that this does indeed satisfy the conditions for a P-structure onj_X:X,→ExX.

∙ This description is useful for proving properties of Ex^∞X, whence there is an elementary deduction of the Kan-Quillen

Thanks for listening!

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