Spahn
monoidal quasicategory (Rev #8, 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 §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).

Reference

  • DAGII

Revision on February 11, 2013 at 23:19:36 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.