Spahn monoidal quasicategory (Rev #2)

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(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 $\tilde{C^\times}$ and 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 $f(\{i,\dots,j\})=f(\{i,i+1\})\times \dots\times f(\{j-1,j\})$.

Then $p:C^\times\to N(\Delta)^{op}$ is a monoidal structure iff $C$ admits finite products. Here $p$ is the restriction of the projection $\tilde{C^\times}\to N(\Delta)$.

Monoidal structure for endomorphism algebras

DAGII §2.7

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

Define the category $J\supset \Delta$ by adding intervals (then we have $\Delta^\times$ as above) or the point $*$ to $\Delta$. 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. Define $J_{[n]}:=J\times_\Delta \{[n]\}$ which is either an interval $\i\le j$ in $\Delta[n]$ or $*$. A vertex of $\tilde{C[m]^\otimes}$ can be identified with a functor $f:N(J_{[n]})^{op}\to M^\otimes$ covering the map $N(J_{[n]})\to N(\Delta^\prime)$ induced by $\psi^\prime$.

Define $C[m]^\otimes$ to be the full simplicial subset of $\tilde{C[m]^\otimes}$ spanned by the objects classifying those functors $f:N(J_{[n]})^{op}\to M^\otimes$ which satisfy

(1) $qf(a)\in hom(\Delta^1 ,C^\otimes)$ is $p$-cocartesian for every $a\in J_{[n]}$.

(2) $f(a)$ is $pq$-cocartesian for every $a:([n],*)\to ([n],i\le j)$ corresponding to $j\in \{i,\dots,j\}$.

Finally define $C[m]:=C[m]_{[1]}^\otimes$. Then the above constructed map $C[m]^\otimes\to N(\Delta)^{op}$ is a monoidal category. The restriction to $\Delta^\prime\subseteq J$ induces a monoidal functor $C[m]^\otimes C^\otimes$.

Reference

• DAGII