monoidal quasicategory (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 quasicategory. 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 \to C^\otimes.

The composition monoidal structure for endofunctor algebras, monads (DAGII)

DAGII §3.1

(Notation 3.1.6): Define functors E,E¯:Δ opsSetE,\overline{E}:\Delta^{op}\to sSet by the following:

(1) Let n0n\ge 0, M,KsSetM,K\in sSet. A morphism KE([n])K\to E([n]) is given by a collection (s ijhom K(K×M,K×M) 0ijn(s_{ij}\in hom_K(K\times M,K\times M)_{0\le i\le j\le n} satisfying s ii=ids_{ii}=id and s ijs jk=s iks_{ij} s_{jk}=s_{ik} for 0ijn0\le i\le j\le n.

(2) Let n0n\ge 0, M,KsSetM,K\in sSet. A morphism KE¯([n])K\to \overline{E}([n]) is given by two collection (s ijhom K(K×M,K×M) 0ijn(s_{ij}\in hom_K(K\times M,K\times M)_{0\le i\le j\le n} and (t ihom K(K,K×M) 0in(t_{i}\in hom_K(K,K\times M)_{0\le i\le n} satisfying s ii=ids_{ii}=id, s ijs jk=s iks_{ij} s_{jk}=s_{ik}, and t i=s ijt jt_i=s_ij t_j for 0ijn0\le i\le j\le n.

(3) Morphisms E([n])E([m])E([n])\to E([m]) resp. E¯([n])E¯([m])\overline{E}([n])\to \overline{E}([m]) are induced by composition with a:[m][n]a:[m]\to [n].

(4) Define End (M):=N E(Δ op)End^\otimes(M):=N_E(\Delta^{op}) and End ¯(M):=N E¯(Δ op)\overline{End^\otimes}(M):=N_{\overline{E}}(\Delta^{op}). Here N E(Δ opN_E(\Delta^{op} denotes the relative nerve of Δ op\Delta^{op} relative EE.

(Proposition 3.1.7): Let End [n] (M)End_{[n]}^\otimes(M) denote the fiber of the projection p:End (M)N(Δ op)p:End^\otimes(M)\to N(\Delta^{op}) over [n][n]. Let End¯ [n] (M)\overline{End}_{[n]}^\otimes(M) denote the fiber of the projection q:End¯ (M)N(Δ op)q:\overline{End}^\otimes(M)\to N(\Delta^{op}) over [n][n]. Then in

End¯ (M)N(Δ op)qEnd [n] (M)pN(Δ op)\overline{End}^\otimes(M)\to N(\Delta^{op})\stackrel{q}{\to}End_{[n]}^\otimes(M)\stackrel{p}{\to}N(\Delta^{op})

we have that:

(1) pp and pqpq are cocartesian fibrations and qq is a categorical fibration.

(2) End ¯ [n](M)Fun(M,M) n×M\overline{End^{\otimes}}_{[n]}(M)\simeq Fun(M,M)^n\times M and End [n] (M)M nEnd^\otimes_{[n]}(M)\simeq M^n.

(3) The restriction of the above diagram

MFun(M,M)N(Δ op)M\to Fun(M,M)\to N(\Delta^{op})

exhibits MM as left tensored over Fun(M,M)Fun(M,M) and Fun(M,M)Fun(M,M) as a monoidal quasicategory. This monoidal structure is called composition monoidal structure.

(Definition 3.1.8): A monad on a quasicategory MM is defined to be an algebra object in the composition monoidal quasicategory Fun(M,M)Fun(M,M).

The composition monoidal structure for endofunctor algebras, monads (Higher Algebra)

In the language of (,1)(\infty,1)-operads the above description reads as follows:

(Higher Algebra, Definition LetbfLML \bf{L} {LM} denote the colored operad defined by:


(1) LM{LM} has two objects aa and mm.

(2) Mul LM({X i} iI,a)Mul_{LM}(\{X_i\}_{i\in I},a) is the collection of all linear orderings of the set II.

Mul LM({X i} iI,m)Mul_{LM}(\{X_i\}_{i\in I},m) is the collection of all linear orderings {i 1<<i n}\{i_1\lt\dots\lt i_n\} of the set II such that X i n=mX_{i_n}=m and X i j=aX_{i_j}=a for j<nj\lt n; if II is empty also Mul LM({X i} iI,m)Mul_{LM}(\{X_i\}_{i\in I},m) shall be empty.

(3) The composition law on LM{LM} shall be determined by composition of linear orderings (Definition

(Remark There is a unique operation ϕMul LM({a,m},m)\phi\in Mul_{LM}(\{a,m\},m). If CC is a symmetric monoidal category and F:LMCF:LM\to C is a map of colored operads, then we can identify the restriction F|Ass:AssCF| Ass:Ass\to C with an associative algebra object A:=F(a)A:=F(a) in CC. In this case F(ϕ):F(a)F(m)F(a)F(\phi):F(a)\otimes F(m)\to F(a) exhibits F(m)F(m) as a left F(a)F(a)-module.

(Remark The restriction of LM{LM} to the object aa is a sub-colered operad of LM{LM} which is isomorphic to the associative operad AssAss.(…)

(Higher Algebra, Proposition


  • Jacob Lurie, DAGII

  • Jacob Lurie, Higher Algebra

Last revised on February 12, 2013 at 06:00:40. See the history of this page for a list of all contributions to it.