Spahn
monoidal quasicategory (Rev #1, 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} extract C ×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\}).

Monoidal structure for endomorphism algebras

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

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]).

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

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

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

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

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..

Reference

  • DAGII

Revision on February 10, 2013 at 03:17:29 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.