Spahn monoidal quasicategory (Rev #1)

(…)

(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

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

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

where the vertical morphisms of the top square are inclusions..

Reference

• DAGII