(...) (M1) $p:C^\otimes\to \Delta^{op}$ cocartesian fibration. (M2) $C^\otimes_{[n]}\simeq C^n$. ## Constructions of monoidal structures ### Monoidal structure for a quasicategory with finite products DAGII § 1.2 Idea: Take as $n$-sequences $n$-fold products to obtain $\tilde{C^\times}$ extract $C^\times$ form $\tilde{C^\times}$ via (M2). Construction: Add intervals to $\Delta$: Let $\Delta^\times$ have as objects pairs $([n],i\le j)$ where $0\le i\le j\le n$. Define $\tilde{C^\times}$ by $$hom(K\times_{N(\Delta)^{op}} N(\Delta^\times)^{op}, C)=:hom(K,\tilde{C^\times}).$$ Denote the fiber over $[n]$ of $\tilde{C^\times}$ by $\tilde{C^\times}_{[n]}$. Denote the poset of intervals in $[n]$ by $P_n$. The we have $\tilde{C^\times}_{[n]}=Fun(N(P_n)^{op}, C)$. Let $C^\times$ denote the full simplicial subset on those functors $f(\{i,i+1,\dots,j\})\to f(\{k,k+1\}) $ entailing $f(\{i,\dots,j\})=f(\{i,i+1\})\times \dots\times f(\{j-1,j\})$. ### Monoidal structure for endomorphism algebras Define the category $J\supset \Delta$ by adding intervals (then we have $\Delta^\times$ as above) of the point $*$. More precisely: An object of $J$ is a pair $([n],i\le j)$ or $([n],*)$. Morphisms are "narrowings": a morphism $a:([m],i\le j)\to ([n],i^\prime\le j^\prime)$ is a morphism $\underline{a}:[m]\to[n]$ satisfying $i^\prime\le a(i)\le a(j)\le j^\prime$; $hom(([m],i\le j), ([n],*)):=\emptyset$; $hom(([m],*), ([n],i\le j))=\{(a,k),a:[m]\to [n], i\le k\le j\}$; and $hom(([m],*),([n],*))=hom([m],[n])$. $\Delta$ can be identified with two different subcategories of $J$. Define $$\psi:\begin{cases}J\to \Delta\\([n],i\le j)\mapsto [n]\end{cases}$$ $$\psi^\prime:\begin{cases}J\to \Delta^\prime\\([n],i\le j)\mapsto {i,i+1,\dots,j}\\([n],*)\mapsto [0].\end{cases}$$ where $\Delta^\prime=\Delta$ are considered as subcategories of $J$ in different ways as indicated. Let $m\in M$ be an object. The category $\tilde{C[m]^\otimes}$ equipped with a map $\tilde{C[m]^\otimes}\to N(\Delta^{op})$ is defined by $hom_{N(\Delta)^{op})}(K,\tilde{C[m]^\otimes})$ being in bijection with diagrams of type $$\array{ K\times_{N(\Delta)^{op}}N(\Delta)^{op}&\to&\{m\}\\ \downarrow&&\downarrow\\ K\times_{N(\Delta)^{op}}N(J)^{op}&\to&M\\ \downarrow&&\downarrow\\ N(\Delta^\prime)^{op}&\stackrel{id}{\to}& N(\Delta^\prime)^{op} }$$ where the vertical morphisms of the top square are inclusions.. ## Reference * DAGII