Spahn monoidal quasicategory

(…)

(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

$$hom(K\times_{N(\Delta)^{op}} N(\Delta^\times)^{op}, C)=:hom(K,\tilde{C^\times}).$$

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

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 quasicategory. 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

$$\psi:\begin{cases}J\to \Delta\\([n],i\le j)\mapsto [n]\end{cases}$$

$$\psi^\prime:\begin{cases}J\to \Delta^\prime\\([n],i\le j)\mapsto \{i,i+1,\dots,j\}\\([n],*)\mapsto [0].\end{cases}$$

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. 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 $C[m]^\otimes \to C^\otimes$.

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

DAGII §3.1

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

(1) Let $n\ge 0$, $M,K\in sSet$. A morphism $K\to E([n])$ is given by a collection $(s_{ij}\in hom_K(K\times M,K\times M)_{0\le i\le j\le n}$ satisfying $s_{ii}=id$ and $s_{ij} s_{jk}=s_{ik}$ for $0\le i\le j\le n$.

(2) Let $n\ge 0$, $M,K\in sSet$. A morphism $K\to \overline{E}([n])$ is given by two collection $(s_{ij}\in hom_K(K\times M,K\times M)_{0\le i\le j\le n}$ and $(t_{i}\in hom_K(K,K\times M)_{0\le i\le n}$ satisfying $s_{ii}=id$, $s_{ij} s_{jk}=s_{ik}$, and $t_i=s_ij t_j$ for $0\le i\le j\le n$.

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

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

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

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

we have that:

(1) $p$ and $pq$ are cocartesian fibrations and $q$ is a categorical fibration.

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

(3) The restriction of the above diagram

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

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

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

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

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

(Higher Algebra, Definition 4.2.1.1) Let ${LM}$ denote the colored operad defined by:

(1) ${LM}$ has two objects $a$ and $m$.

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

$Mul_{LM}(\{X_i\}_{i\in I},m)$ is the collection of all linear orderings $\{i_1\lt\dots\lt i_n\}$ of the set $I$ such that $X_{i_n}=m$ and $X_{i_j}=a$ for $j\lt n$; if $I$ is empty also $Mul_{LM}(\{X_i\}_{i\in I},m)$ shall be empty.

(3) The composition law on ${LM}$ shall be determined by composition of linear orderings (Definition 4.1.1.1).

(Remark 4.2.1.3) There is a unique operation $\phi\in Mul_{LM}(\{a,m\},m)$. If $C$ is a symmetric monoidal category and $F:LM\to C$ is a map of colored operads, then we can identify the restriction $F| Ass:Ass\to C$ with an associative algebra object $A:=F(a)$ in $C$. In this case $F(\phi):F(a)\otimes F(m)\to F(a)$ exhibits $F(m)$ as a left $F(a)$-module.

(Remark 4.2.1.4) The restriction of ${LM}$ to the object $a$ is a sub-colered operad of ${LM}$ which is isomorphic to the associative operad $Ass$.(…)

(Higher Algebra, Proposition 6.2.0.2)

Reference

• Jacob Lurie, DAGII

• Jacob Lurie, Higher Algebra