Spahn
monoidal quasicategory (Rev #2, changes)

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(M1) p:C Δ opp:C^\otimes\to \Delta^{op} cocartesian fibration.

(M2) C [n] C nC^\otimes_{[n]}\simeq C^n.

Constructions of monoidal structures

Monoidal structure for a quasicategory with finite products

DAGII § 1.2

Idea: Take as nn-sequences nn-fold products to obtain C ט\tilde{C^\times} and extractC ×C^\times form C ט\tilde{C^\times} via (M2).

Construction: Add intervals to Δ\Delta: Let Δ ×\Delta^\times have as objects pairs ([n],ij)([n],i\le j) where 0ijn0\le i\le j\le n. Define C ט\tilde{C^\times} by

hom(K× N(Δ) opN(Δ ×) op,C)=:hom(K,C ט).hom(K\times_{N(\Delta)^{op}} N(\Delta^\times)^{op}, C)=:hom(K,\tilde{C^\times}).

Denote the fiber over [n][n] of C ט\tilde{C^\times} by C ט [n]\tilde{C^\times}_{[n]}. Denote the poset of intervals in [n][n] by P nP_n. The we have C ט [n]=Fun(N(P n) op,C)\tilde{C^\times}_{[n]}=Fun(N(P_n)^{op}, C). Let C ×C^\times denote the full simplicial subset on those functors f({i,i+1,,j})f({k,k+1})f(\{i,i+1,\dots,j\})\to f(\{k,k+1\}) entailing f({i,,j})=f({i,i+1})××f({j1,j})f(\{i,\dots,j\})=f(\{i,i+1\})\times \dots\times f(\{j-1,j\}).

Then p:C ×N(Δ) opp:C^\times\to N(\Delta)^{op} is a monoidal structure iff CC admits finite products. Here pp is the restriction of the projection C טN(Δ)\tilde{C^\times}\to N(\Delta).

Monoidal structure for endomorphism algebras

Define DAGII the §2.7 categoryJΔJ\supset \Delta by adding intervals (then we have Δ ×\Delta^\times as above) of the point **. More precisely:

An The object purpose of the following construction is to realize an endomorphism object J End(m) J End(m) is as a an pair algebra object in some category. More precisely we will haveEnd(m)=*Alg(C[ n m],ij) ([n],i\le End(m)=* j) \in Alg(C[m]) or is the terminal object inAlg(C[ n m],*) ([n],*) Alg(C[m]) . Morphisms So are “narrowings”: a morphism a End:([m],ij)([n],i j ) a:([m],i\le End(m) j)\to ([n],i^\prime\le j^\prime) is a “universal” morphism among all objects acting ona̲:[m][n] \underline{a}:[m]\to[n] m satisfying i a(i)a(j)j i^\prime\le a(i)\le a(j)\le j^\prime; hom(([m],ij),([n],*)):=hom(([m],i\le j), ([n],*)):=\emptyset; hom(([m],*),([n],ij))={(a,k),a:[m][n],ikj}hom(([m],*), ([n],i\le j))=\{(a,k),a:[m]\to [n], i\le k\le j\}; and hom(([m],*),([n],*))=hom([m],[n])hom(([m],*),([n],*))=hom([m],[n]).

Δ\DeltaDefine the category can be identified with two different subcategories of JΔJ\supset \DeltaJJ by adding intervals (then we have . DefineΔ ×\Delta^\times as above) or the point ** to Δ\Delta. More precisely:

ψ:{JΔ ([n],ij)[n]\psi:\begin{cases}J\to \Delta\\([n],i\le j)\mapsto [n]\end{cases}

An object of JJ is a pair ([n],ij)([n],i\le j) or ([n],*)([n],*). Morphisms are “narrowings”: a morphism a:([m],ij)([n],i j )a:([m],i\le j)\to ([n],i^\prime\le j^\prime) is a morphism a̲:[m][n]\underline{a}:[m]\to[n] satisfying i a(i)a(j)j i^\prime\le a(i)\le a(j)\le j^\prime; hom(([m],ij),([n],*)):=hom(([m],i\le j), ([n],*)):=\emptyset; hom(([m],*),([n],ij))={(a,k),a:[m][n],ikj}hom(([m],*), ([n],i\le j))=\{(a,k),a:[m]\to [n], i\le k\le j\}; and hom(([m],*),([n],*))=hom([m],[n])hom(([m],*),([n],*))=hom([m],[n]).

ψ :{JΔ ([n],ij)i,i+1,,j ([n],*)[0].\psi^\prime:\begin{cases}J\to \Delta^\prime\\([n],i\le j)\mapsto {i,i+1,\dots,j}\\([n],*)\mapsto [0].\end{cases}

Δ\Delta can be identified with two different subcategories of JJ. Define

where Δ =Δ\Delta^\prime=\Delta are considered as subcategories of JJ in different ways as indicated.

ψ:{JΔ ([n],ij)[n]\psi:\begin{cases}J\to \Delta\\([n],i\le j)\mapsto [n]\end{cases}

Let mMm\in M be an object. The category C[m] ˜\tilde{C[m]^\otimes} equipped with a map C[m] ˜N(Δ op)\tilde{C[m]^\otimes}\to N(\Delta^{op}) is defined by hom N(Δ) op)(K,C[m] ˜)hom_{N(\Delta)^{op})}(K,\tilde{C[m]^\otimes}) being in bijection with diagrams of type

ψ :{JΔ ([n],ij){i,i+1,,j} ([n],*)[0].\psi^\prime:\begin{cases}J\to \Delta^\prime\\([n],i\le j)\mapsto \{i,i+1,\dots,j\}\\([n],*)\mapsto [0].\end{cases}
K× N(Δ) opN(Δ) op {m} K× N(Δ) opN(J) op M N(Δ ) op id N(Δ ) op\array{ K\times_{N(\Delta)^{op}}N(\Delta)^{op}&\to&\{m\}\\ \downarrow&&\downarrow\\ K\times_{N(\Delta)^{op}}N(J)^{op}&\to&M\\ \downarrow&&\downarrow\\ N(\Delta^\prime)^{op}&\stackrel{id}{\to}& N(\Delta^\prime)^{op} }

where Δ =Δ\Delta^\prime=\Delta are considered as subcategories of JJ in different ways as indicated.

where Let the vertical morphisms of the top square are inclusions..mMm\in M be an object. The category C[m] ˜\tilde{C[m]^\otimes} equipped with a map C[m] ˜N(Δ op)\tilde{C[m]^\otimes}\to N(\Delta^{op}) is defined by hom N(Δ) op)(K,C[m] ˜)hom_{N(\Delta)^{op})}(K,\tilde{C[m]^\otimes}) being in bijection with diagrams of type

K× N(Δ) opN(Δ) op {m} K× N(Δ) opN(J) op M N(Δ ) op id N(Δ ) op\array{ K\times_{N(\Delta)^{op}}N(\Delta)^{op}&\to&\{m\}\\ \downarrow&&\downarrow\\ K\times_{N(\Delta)^{op}}N(J)^{op}&\to&M\\ \downarrow&&\downarrow\\ N(\Delta^\prime)^{op}&\stackrel{id}{\to}& N(\Delta^\prime)^{op} }

where the vertical morphisms of the top square are inclusions. Define J [n]:=J× Δ{[n]}J_{[n]}:=J\times_\Delta \{[n]\} which is either an interval ij\i\le j in Δ[n]\Delta[n] or **. A vertex of C[m] ˜\tilde{C[m]^\otimes} can be identified with a functor f:N(J [n]) opM f:N(J_{[n]})^{op}\to M^\otimes covering the map N(J [n])N(Δ )N(J_{[n]})\to N(\Delta^\prime) induced by ψ \psi^\prime.

Define C[m] C[m]^\otimes to be the full simplicial subset of C[m] ˜\tilde{C[m]^\otimes} spanned by the objects classifying those functors f:N(J [n]) opM f:N(J_{[n]})^{op}\to M^\otimes which satisfy

(1) qf(a)hom(Δ 1,C )qf(a)\in hom(\Delta^1 ,C^\otimes) is pp-cocartesian for every aJ [n]a\in J_{[n]}.

(2) f(a)f(a) is pqpq-cocartesian for every a:([n],*)([n],ij)a:([n],*)\to ([n],i\le j) corresponding to j{i,,j}j\in \{i,\dots,j\}.

Finally define C[m]:=C[m] [1] C[m]:=C[m]_{[1]}^\otimes. Then the above constructed map C[m] N(Δ) opC[m]^\otimes\to N(\Delta)^{op} is a monoidal category. The restriction to Δ J\Delta^\prime\subseteq J induces a monoidal functor C[m] C C[m]^\otimes C^\otimes.

Reference

  • DAGII

Revision on February 10, 2013 at 06:35:04 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.