Join (simplicial Sets)
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In
higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
in
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the join of simplicial sets is an operation making the category of simplicial sets into a
monoidal category In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (cate ...
. 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 A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...
construction, it corresponds to the join of categories and under the geometric realization, it corresponds to the join of topological spaces.


Definition

For natural numbers m,p,q\in\mathbb, one has the identity:Cisinski 2019, 3.4.12. : \operatorname( +q+1 =\prod_\operatorname( \times\operatorname( , which can be extended by colimits to a functor a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
-*-\colon \mathbf\times\mathbf\rightarrow \mathbf, which together with the empty simplicial set as unit element makes the category of simplicial sets \mathbf into a
monoidal category In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an Object (cate ...
. For simplicial set X and Y, their ''join'' X*Y is the simplicial set: : (X*Y)_n =\prod_X_i\times Y_j. A n-simplex \sigma\colon \Delta^n\rightarrow X*Y therefore either factors over X or Y or splits into a p-simplex \sigma_-\colon \Delta^p\rightarrow X and a q-simplex \sigma_+\colon \Delta^q\rightarrow Y with n=p+q+1 and \sigma=\sigma_-*\sigma_+. One has canonical morphisms X,Y\rightarrow X*Y, which combine into a canonical morphism X+Y\rightarrow X*Y through the
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
of the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...
. One also has a canonical morphism X*Y\rightarrow\Delta^0*\Delta^0 \cong\Delta^1 of terminal maps, for which the fiber of 0 is X and the fiber of 1 is Y. For a simplicial set X, one further defines its ''left cone'' and ''right cone'' as: : X^\triangleleft :=\Delta^0*X, : X^\triangleright :=X*\Delta^0.


Right adjoint

Let Y be a simplicial set. The functor Y*-\colon \mathbf\rightarrow Y\backslash\mathbf, X\mapsto(Y\mapsto Y*X) has a
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
Y\backslash\mathbf\rightarrow\mathbf, (t\colon Y\rightarrow W)\mapsto t\backslash W (alternatively denoted Y\backslash W) and the functor -*Y\colon \mathbf\rightarrow Y\backslash\mathbf, X\mapsto(Y\mapsto X*Y) also has a right adjoint Y\backslash\mathbf\rightarrow\mathbf, (t\colon Y\rightarrow W)\mapsto W/t (alternatively denoted W/Y).Lurie 2009, Proposition 1.2.9.2Cisinski 2019, 3.4.14. A special case is Y=\Delta^0 the terminal simplicial set, since \mathbf_* =\Delta^0\backslash\mathbf is the category of pointed simplicial sets. Let \mathcal be a category and X\in\operatorname\mathcal be an object. Let /math> be the terminal category (with the notation taken from the
terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
of the simplex category), then there is an associated functor t\colon rightarrow\mathcal, 0\mapsto X, which with the
nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...
induces a morphism Nt\colon \Delta^0\rightarrow N\mathcal. For every simplicial set A, one has by additionally using the adjunction between the join of categories and slice categories: : \begin \mathbf(A,N\mathcal/Nt) &\cong\mathbf_*(\Delta^0\rightarrow A*\Delta^0,Nt) \cong\mathbf_*( rightarrow\tau(A)\star t) \\ &\cong\mathbf(\tau(A),\mathcal/X) \cong\mathbf(A,N(\mathcal/X)). \end Hence according to the
Yoneda lemma In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
, one has (with the alternative notation, which here better underlines the result): : N\mathcal/NX \cong N(\mathcal/X).


Examples

One has: : \partial\Delta^m*\Delta^n\cup\Delta^m*\partial\Delta^n \cong\partial\Delta^, : \Lambda_k^m*\Delta^n\cup\Delta^m*\partial\Delta^n \cong\Lambda_k^, : \partial\Delta^m*\Delta^n\cup\Delta^m*\Lambda_k^n \cong\Lambda_^.


Properties

* For simplicial sets X and Y, there is a unique morphism \gamma_\colon X\diamond Y\rightarrow X*Y into the diamond operation compatible with the maps X+Y\rightarrow X*Y,X\diamond Y and X*Y,X\diamond Y\rightarrow\Delta^1.Cisinski 2019, Proposition 4.2.2. It is a weak categorical equivalence, hence a weak equivalence of the Joyal model structure.Lurie 2009, Proposition 4.2.1.2.Cisinksi 2019, Proposition 4.2.3. * For a simplicial set X, the functors X*-,-*X\colon\mathbf\rightarrow\mathbf preserve weak categorical equivalences. * For ∞-categories X and Y, the simplicial set X*Y is also an ∞-category. * The join is associative. For simplicial sets X, Y and Z, one has: *: (X*Y)*Z \cong X*(Y*Z). * The join reverses under the opposite simplicial set. For simplicial sets X and Y, one has:Joyal 2008, p. 244Cisinski 2019, Remark 3.4.15. *: (X*Y)^\mathrm \cong Y^\mathrm*X^\mathrm. * For a morphism t\colon Y\rightarrow W, one has (as adjoint of the previous result): *: (W/t)^\mathrm \cong t^\mathrm\backslash W^\mathrm. * For morphisms z\colon Y*X\rightarrow W, its precomposition with the canonical inclusion x\colon X\rightarrow Y*X\rightarrow W and y\colon Y\rightarrow W/x, one has W/z\cong(W/x)/y or in alternative notation: :: W/(Y*X) \cong(W/X)/Y. : For every simplicial set A, one has: :: \begin \mathbf(A,W/z) &\cong(Y*X)\backslash\mathbf((Y*X)\rightarrow A*(Y*X),z) \cong X\backslash\mathbf(X\rightarrow(A*Y)*X,x) \\ &\cong\mathbf(A*Y,W/x) \cong Y\backslash\mathbf(Y\rightarrow A*Y,y) \cong\mathbf(A,(W/x)/y), \end : so the claim follows from the Yoneda lemma. * Under the
nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...
, the join of categories becomes the join of simplicial sets. For small categories \mathcal and \mathcal, one has:Joyal 2008, Corollary 3.3.Kerodon
Example 4.3.3.14.
/ref> *: N(\mathcal\star\mathcal) \cong N\mathcal*N\mathcal.


Literature

* * * {{cite book , last=Cisinski , first=Denis-Charles , author-link=Denis-Charles Cisinski , url=https://cisinski.app.uni-regensburg.de/CatLR.pdf , title=Higher Categories and Homotopical Algebra , date=2019-06-30 , publisher=
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, isbn=978-1108473200 , location= , language=en , authorlink=


References


External links

* join of simplicial sets and join of quasi-categories at the ''n''Lab
Joins of Simplicial Sets
on Kerodon Higher category theory Simplicial sets