In higher category theory in mathematics , the join of simplicial sets is an operation making the category of simplicial sets into a monoidal category . In particular, it takes two simplicial sets to construct another simplicial set. It is closely related to the diamond operation and used in the construction of the twisted diagonal . Under the nerve construction, it corresponds to the join of categories and under the geometric realization , it corresponds to the join of topological spaces .
Definition Visualization of the join X ∗ Y {\displaystyle X*Y} with the blue part representing X {\displaystyle X} and the green part representing Y {\displaystyle Y} . For natural numbers m , p , q ∈ N {\displaystyle m,p,q\in \mathbb {N} } , one has the identity:[ 1]
Hom ( [ m ] , [ p + q + 1 ] ) = ∏ i + j + 1 = n Hom ( [ i ] , [ p ] ) × Hom ( [ j ] , [ q ] ) , {\displaystyle \operatorname {Hom} ([m],[p+q+1])=\prod _{i+j+1=n}\operatorname {Hom} ([i],[p])\times \operatorname {Hom} ([j],[q]),} which can be extended by colimits to a functor a functor − ∗ − : s S e t × s S e t → s S e t {\displaystyle -*-\colon \mathbf {sSet} \times \mathbf {sSet} \rightarrow \mathbf {sSet} } , which together with the empty simplicial set as unit element makes the category of simplicial sets s S e t {\displaystyle \mathbf {sSet} } into a monoidal category . For simplicial set X {\displaystyle X} and Y {\displaystyle Y} , their join X ∗ Y {\displaystyle X*Y} is the simplicial set:[ 2] [ 3] [ 1]
( X ∗ Y ) n = ∏ i + j + 1 = n X i × Y j . {\displaystyle (X*Y)_{n}=\prod _{i+j+1=n}X_{i}\times Y_{j}.} A n {\displaystyle n} -simplex σ : Δ n → X ∗ Y {\displaystyle \sigma \colon \Delta ^{n}\rightarrow X*Y} therefore either factors over X {\displaystyle X} or Y {\displaystyle Y} or splits into a p {\displaystyle p} -simplex σ − : Δ p → X {\displaystyle \sigma _{-}\colon \Delta ^{p}\rightarrow X} and a q {\displaystyle q} -simplex σ + : Δ q → Y {\displaystyle \sigma _{+}\colon \Delta ^{q}\rightarrow Y} with n = p + q + 1 {\displaystyle n=p+q+1} and σ = σ − ∗ σ + {\displaystyle \sigma =\sigma _{-}*\sigma _{+}} .[ 4]
One has canonical morphisms X , Y → X ∗ Y {\displaystyle X,Y\rightarrow X*Y} , which combine into a canonical morphism X + Y → X ∗ Y {\displaystyle X+Y\rightarrow X*Y} through the universal property of the coproduct . One also has a canonical morphism X ∗ Y → Δ 0 ∗ Δ 0 ≅ Δ 1 {\displaystyle X*Y\rightarrow \Delta ^{0}*\Delta ^{0}\cong \Delta ^{1}} of terminal maps, for which the fiber of 0 {\displaystyle 0} is X {\displaystyle X} and the fiber of 1 {\displaystyle 1} is Y {\displaystyle Y} .
For a simplicial set X {\displaystyle X} , one further defines its left cone and right cone as:
X ◃ := Δ 0 ∗ X , {\displaystyle X^{\triangleleft }:=\Delta ^{0}*X,} X ▹ := X ∗ Δ 0 . {\displaystyle X^{\triangleright }:=X*\Delta ^{0}.}
Right adjoint Let Y {\displaystyle Y} be a simplicial set. The functor Y ∗ − : s S e t → Y ∖ s S e t , X ↦ ( Y ↦ Y ∗ X ) {\displaystyle Y*-\colon \mathbf {sSet} \rightarrow Y\backslash \mathbf {sSet} ,X\mapsto (Y\mapsto Y*X)} has a right adjoint Y ∖ s S e t → s S e t , ( t : Y → W ) ↦ t ∖ W {\displaystyle Y\backslash \mathbf {sSet} \rightarrow \mathbf {sSet} ,(t\colon Y\rightarrow W)\mapsto t\backslash W} (alternatively denoted Y ∖ W {\displaystyle Y\backslash W} ) and the functor − ∗ Y : s S e t → Y ∖ s S e t , X ↦ ( Y ↦ X ∗ Y ) {\displaystyle -*Y\colon \mathbf {sSet} \rightarrow Y\backslash \mathbf {sSet} ,X\mapsto (Y\mapsto X*Y)} also has a right adjoint Y ∖ s S e t → s S e t , ( t : Y → W ) ↦ W / t {\displaystyle Y\backslash \mathbf {sSet} \rightarrow \mathbf {sSet} ,(t\colon Y\rightarrow W)\mapsto W/t} (alternatively denoted W / Y {\displaystyle W/Y} ).[ 5] [ 6] [ 7] A special case is Y = Δ 0 {\displaystyle Y=\Delta ^{0}} the terminal simplicial set, since s S e t ∗ = Δ 0 ∖ s S e t {\displaystyle \mathbf {sSet} _{*}=\Delta ^{0}\backslash \mathbf {sSet} } is the category of pointed simplicial sets.
Let C {\displaystyle {\mathcal {C}}} be a category and X ∈ Ob C {\displaystyle X\in \operatorname {Ob} {\mathcal {C}}} be an object. Let [ 0 ] {\displaystyle [0]} be the terminal category (with the notation taken from the terminal object of the simplex category ), then there is an associated functor t : [ 0 ] → C , 0 ↦ X {\displaystyle t\colon [0]\rightarrow {\mathcal {C}},0\mapsto X} , which with the nerve induces a morphism N t : Δ 0 → N C {\displaystyle Nt\colon \Delta ^{0}\rightarrow N{\mathcal {C}}} . For every simplicial set A {\displaystyle A} , one has by additionally using the adjunction between the join of categories and slice categories:[ 8]
s S e t ( A , N C / N t ) ≅ s S e t ∗ ( Δ 0 → A ∗ Δ 0 , N t ) ≅ C a t ∗ ( [ 0 ] → τ ( A ) ⋆ [ 0 ] , t ) ≅ C a t ( τ ( A ) , C / X ) ≅ s S e t ( A , N ( C / X ) ) . {\displaystyle {\begin{aligned}\mathbf {sSet} (A,N{\mathcal {C}}/Nt)&\cong \mathbf {sSet} _{*}(\Delta ^{0}\rightarrow A*\Delta ^{0},Nt)\cong \mathbf {Cat} _{*}([0]\rightarrow \tau (A)\star [0],t)\\&\cong \mathbf {Cat} (\tau (A),{\mathcal {C}}/X)\cong \mathbf {sSet} (A,N({\mathcal {C}}/X)).\end{aligned}}} Hence according to the Yoneda lemma , one has (with the alternative notation, which here better underlines the result):[ 9] [ 7]
N C / N X ≅ N ( C / X ) . {\displaystyle N{\mathcal {C}}/NX\cong N({\mathcal {C}}/X).}
Examples One has:[ 10]
∂ Δ m ∗ Δ n ∪ Δ m ∗ ∂ Δ n ≅ ∂ Δ m + n + 1 , {\displaystyle \partial \Delta ^{m}*\Delta ^{n}\cup \Delta ^{m}*\partial \Delta ^{n}\cong \partial \Delta ^{m+n+1},} Λ k m ∗ Δ n ∪ Δ m ∗ ∂ Δ n ≅ Λ k m + n + 1 , {\displaystyle \Lambda _{k}^{m}*\Delta ^{n}\cup \Delta ^{m}*\partial \Delta ^{n}\cong \Lambda _{k}^{m+n+1},} ∂ Δ m ∗ Δ n ∪ Δ m ∗ Λ k n ≅ Λ m + k + 1 m + n + 1 . {\displaystyle \partial \Delta ^{m}*\Delta ^{n}\cup \Delta ^{m}*\Lambda _{k}^{n}\cong \Lambda _{m+k+1}^{m+n+1}.}
Properties For simplicial sets X {\displaystyle X} and Y {\displaystyle Y} , there is a unique morphism γ X , Y : X ⋄ Y → X ∗ Y {\displaystyle \gamma _{X,Y}\colon X\diamond Y\rightarrow X*Y} into the diamond operation compatible with the maps X + Y → X ∗ Y , X ⋄ Y {\displaystyle X+Y\rightarrow X*Y,X\diamond Y} and X ∗ Y , X ⋄ Y → Δ 1 {\displaystyle X*Y,X\diamond Y\rightarrow \Delta ^{1}} .[ 11] It is a weak categorical equivalence, hence a weak equivalence of the Joyal model structure .[ 12] [ 13] For a simplicial set X {\displaystyle X} , the functors X ∗ − , − ∗ X : s S e t → s S e t {\displaystyle X*-,-*X\colon \mathbf {sSet} \rightarrow \mathbf {sSet} } preserve weak categorical equivalences.[ 14] For ∞-categories X {\displaystyle X} and Y {\displaystyle Y} , the simplicial set X ∗ Y {\displaystyle X*Y} is also an ∞-category.[ 15] [ 16] The join is associative. For simplicial sets X {\displaystyle X} , Y {\displaystyle Y} and Z {\displaystyle Z} , one has: ( X ∗ Y ) ∗ Z ≅ X ∗ ( Y ∗ Z ) . {\displaystyle (X*Y)*Z\cong X*(Y*Z).} The join reverses under the opposite simplicial set . For simplicial sets X {\displaystyle X} and Y {\displaystyle Y} , one has:[ 17] [ 18] ( X ∗ Y ) o p ≅ Y o p ∗ X o p . {\displaystyle (X*Y)^{\mathrm {op} }\cong Y^{\mathrm {op} }*X^{\mathrm {op} }.} For a morphism t : Y → W {\displaystyle t\colon Y\rightarrow W} , one has (as adjoint of the previous result):[ 18] ( W / t ) o p ≅ t o p ∖ W o p . {\displaystyle (W/t)^{\mathrm {op} }\cong t^{\mathrm {op} }\backslash W^{\mathrm {op} }.} For a morphism z : Y ∗ X → W {\displaystyle z\colon Y*X\rightarrow W} , its precomposition with the canonical inclusion x : X → Y ∗ X → W {\displaystyle x\colon X\rightarrow Y*X\rightarrow W} and y : Y → W / x {\displaystyle y\colon Y\rightarrow W/x} corresponding to z {\displaystyle z} under the adjunction s S e t ( Y , W / x ) ≅ X ∖ s S e t ( X → Y ∗ X , x ) {\displaystyle \mathbf {sSet} (Y,W/x)\cong X\backslash \mathbf {sSet} (X\rightarrow Y*X,x)} , one has W / z ≅ ( W / x ) / y {\displaystyle W/z\cong (W/x)/y} or in alternative notation:[ 18] W / ( Y ∗ X ) ≅ ( W / X ) / Y . {\displaystyle W/(Y*X)\cong (W/X)/Y.} Proof: For every simplicial set A {\displaystyle A} , one has: s S e t ( A , W / z ) ≅ ( Y ∗ X ) ∖ s S e t ( ( Y ∗ X ) → A ∗ ( Y ∗ X ) , z ) ≅ X ∖ s S e t ( X → ( A ∗ Y ) ∗ X , x ) ≅ s S e t ( A ∗ Y , W / x ) ≅ Y ∖ s S e t ( Y → A ∗ Y , y ) ≅ s S e t ( A , ( W / x ) / y ) , {\displaystyle {\begin{aligned}\mathbf {sSet} (A,W/z)&\cong (Y*X)\backslash \mathbf {sSet} ((Y*X)\rightarrow A*(Y*X),z)\cong X\backslash \mathbf {sSet} (X\rightarrow (A*Y)*X,x)\\&\cong \mathbf {sSet} (A*Y,W/x)\cong Y\backslash \mathbf {sSet} (Y\rightarrow A*Y,y)\cong \mathbf {sSet} (A,(W/x)/y),\end{aligned}}} so the claim follows from the Yoneda lemma. Under the nerve , the join of categories becomes the join of simplicial sets. For small categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}} , one has:[ 19] [ 20] N ( C ⋆ D ) ≅ N C ∗ N D . {\displaystyle N({\mathcal {C}}\star {\mathcal {D}})\cong N{\mathcal {C}}*N{\mathcal {D}}.}
Literature
References ^ a b Cisinski 2019, 3.4.12. ^ Joyal 2008, Proposition 3.1. ^ Lurie 2009, Definition 1.2.8.1. ^ Kerodon, Remark 4.3.3.17. ^ Joyal 2008, Proposition 3.12. ^ Lurie 2009, Proposition 1.2.9.2 ^ a b Cisinski 2019, 3.4.14. ^ Lurie 2009, 1.2.9 Overcategories and Undercategories ^ Joyal 2008, Proposition 3.13. ^ Cisinski 2019, Proposition 3.4.17. ^ Cisinski 2019, Proposition 4.2.2. ^ Lurie 2009, Proposition 4.2.1.2. ^ Cisinksi 2019, Proposition 4.2.3. ^ Cisinski 2019, Corollary 4.2.5. ^ Joyal 2008, Corollary 3.23. ^ Lurie 2009, Proposition 1.2.8.3 ^ Joyal 2008, p. 244 ^ a b c Cisinski 2019, Remark 3.4.15. ^ Joyal 2008, Corollary 3.3. ^ Kerodon, Example 4.3.3.14.
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