Spahn monoidal quasicategory (Rev #2, changes)

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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

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

~~Define~~ DAGII ~~the~~ §2.7 ~~category~~ ~~$J\supset \Delta$~~ ~~by adding intervals (then we have~~ ~~$\Delta^\times$~~ ~~as above) of the point~~ ~~$*$~~ ~~. More precisely:~~

An The ~~object~~ purpose of the following construction is to realize an endomorphism object $J End (m)$ J End(m) is as a an ~~pair~~ algebra object in some category. More precisely we will have $End (m) = * \in Alg (C [n m], i \leq j)$ ~~([n],i\le~~ End(m)=* j) \in Alg(C[m]) or is the terminal object in $Alg (C [n m], *)$ ~~([n],*)~~ Alg(C[m]) . ~~Morphisms~~ So ~~are~~ ~~“narrowings”:~~ a ~~morphism~~ $a End : ([m], i \leq j) \to ([n], i^{'} \leq j^{'})$ ~~a:([m],i\le~~ End(m) ~~j)\to~~ ~~([n],i^\prime\le~~ ~~j^\prime)~~ is a “universal” ~~morphism~~ among all objects acting on $\underset{̲}{a} : [m] \to [n]$ ~~\underline{a}:[m]\to[n]~~ m ~~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~~ ~~$hom(([m],*),([n],*))=hom([m],[n])$~~ .

~~$\Delta$~~ Define the category ~~can be identified with two different subcategories of~~ $J\supset \Delta$ ~~$J$~~ by adding intervals (then we have ~~. Define~~ $\Delta^\times$ as above) or the point $*$ to $\Delta$ . More precisely:

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

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 $hom(([m],*),([n],*))=hom([m],[n])$ .

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

$\Delta$ 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.~~

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

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

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

$\array{ K\times_{N(\Delta)^{op}}N(\Delta)^{op}&\to&\{m\}\\ \downarrow&&\downarrow\\ K\times_{N(\Delta)^{op}}N(J)^{op}&\to&M\\ \downarrow&&\downarrow\\ N(\Delta^\prime)^{op}&\stackrel{id}{\to}& N(\Delta^\prime)^{op} }$

where $\Delta^\prime=\Delta$ are considered as subcategories of $J$ in different ways as indicated.

~~where~~ Let ~~the~~ ~~vertical~~ ~~morphisms~~ of ~~the~~ ~~top~~ ~~square~~ ~~are~~ ~~inclusions..~~ $m\in M$ 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 $C[m]^\otimes C^\otimes$ .

Reference

DAGII

Revision on February 10, 2013 at 06:35:04 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.