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monoidal quasicategory (Rev #3, 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 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\}).

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

DAGII §2.7

The purpose of the following construction is to realize an endomorphism object End(m)End(m) as an algebra object in some category. More precisely we will have End(m)=*Alg(C[m])End(m)=* \in Alg(C[m]) is the terminal object in Alg(C[m])Alg(C[m]). So End(m)End(m) is “universal” among all objects acting on mm.

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

The composition monoidal structure for endofunctor algebras

DAGII §3.1

Define functors E,E¯:Δ opsSetE,\overline{E}:\Delta^{op}\to sSet by the following:

(1) For n0n\ge 0, M,KsSetM,K\in sSet

Reference

  • DAGII

Revision on February 10, 2013 at 07:36:02 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.