Demazure, lectures on p-divisible groups, III.4, duality of finite Witt groups

[[!redirects III.4, duality of finite Witt groups]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_remark}
###### Remark

Let $m, n\gt 1$ and define $m^{W_n}:=ker(F^m:W_{nk}\to W_{nk})$. Then there are morphisms

$$\array{
m^{W_n}
&\stackrel{t}{\to}&
m^{W_{n+1}}
\\
\downarrow^f&&\downarrow^r
\\
{m-1}^{W_n}
&\stackrel{i}{\hookrightarrow}&
m^{W_n}
}$$

where $i$ is the canonical inclusion, and $r,f,t$ are induced by $R,F,T$ where $F$ is [[Frobenius morphism|Frobenius]], $T$ is [[Witt ring|Verschiebung]] and $R:W\to W_n$ is [[Witt ring|restriction]]. $i$ and $t$ are monomorphisms, $f$ and $r$ are epimorphisms, and for $m^{W_n}$ we have $F=if$ and $V=rt$.
=--

+-- {: .num_defn}
###### Definition and Remark

1. For $R\in M_k$, let $W^\prime(R)$ be the set of all $(\alpha_0,\alpha_1,\dots)\in W_k(R)$ such that $a_n=0$ for large $n$, and $a_n$ nilpotent for all $n$.

1. $W^\prime(R)$ is an ideal in $W_k(R)$.

1. $E(w,t)$ is a polynomial for $w\in W^\prime(R)$.

1. In particular $E(w,1)$ is defined for $w\in W^\prime(R)$, and we have a morphism of groups

$$\tilde E:
\begin{cases}
W^\prime\to \mu_k
\\
w\mapsto E(w,1)
\end{cases}$$

If $x\in W_k(R)$, $y\in W^\prime(R)$ then $xy\in W^\prime(R)$ and $E(xy,1)\in R^*$. (...)

The morphism
$$\begin{cases}
W\times W^\prime\to \mu_k
\\
(x,y)\mapsto E(xy,1)
\end{cases}$$
 is bilinear and hence gives a morphism of groups

$$W^\prime\to D(W_k)$$

which is an isomorphism.
=--

+-- {: .num_defn}
###### Definition
Let

$$\sigma_n:\begin{cases}
W_{nk}\to W_k
\\
(\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots)
\end{cases}$$
be the section of $R_n:W_k\to W_{nk}$.

$\sigma_n$ is not a morphism of groups. $\sigma_n$ sends $m^{W_n}$ in $W^\prime$.
=--

+-- {: .num_theorem}
###### Theorem
For $x\in m^{W_n}(R)$, $y\in n^{W_m}(R)$, define

$$\lt x,y\gt:=E(\sigma_n(x)\sigma_m(y),1)$$

then $\lt x,y\gt$ is bilinear and gives an isomorphism

$$m^{W_n}\simeq D(n^{W_m})$$

and satisfies

$$\lt x,t y\gt=\lt f x,y\gt$$

$$\lt x,r y\gt=\lt i x,y\gt$$
=--


Demazure, lectures on p-divisible groups, III.5, Dieudonné modules (affine unipotent groups)

[[!redirects III.5, Dieudonné modules (affine unipotent groups)]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $k$ be a perfect field of prime characteristic $p$.

+-- {: .num_defn}
###### Definition
Let $\underline W$ denote the codirected system of [[unipotent k-group|affine commutative unipotent]] $Acu_k$ $k$-groups

$$W_{1k}\stackrel{T}{\to}W_{2k}\stackrel{T}{\to}W_{3k}\stackrel{T}{\to}\cdots$$

The [[Witt vectors|Witt ring]] $W(k)$ operates on $\underline W$ as follows:

Let $\sigma$ denote the Frobenius morphism $W(k)\to W(k)$, let $a\mapsto a^{(p^n)}$. This Frobenius is bijective since $k$ is perfect. Let $a\in W(k)$, let $w\in W_n(R)$, $R\in M_k$. We define

$$a * w:=a^{p^{1-n}} R\cdot w$$

where $a^{p^{1-n} }R$ is the image of $a^{(p^{-n})}$ in $W(R)$, and $b\cdot w\in W_n(R)$, the product of $b\in W(R)$ and $w\in W_n(R)=W(R)/T^n W(R)$. By this definition $W_n(R)$ becomes a $W(k)$-module, and $T:W_n(R)\to W_{n+1}(R)$ is a homomorphism of $W(k)$-modules since we have

$$T(a* w)= T(a^{(p^{1-n)}} R\cdot w)=T(F(a^{(p^{-n})})R)\cdot w)=a^{p^{-n}}\cdot T w=a * Tw$$

For any $G\in Ac u_k$ the **[[Dieudonné module]] $M(G)$ of $G$ is defined to be the $W(k)$-module

$$M(G)=colim_n Acu_k(G,W_{nk})$$

or- equivalently- $M(G)=codir(Acu_k)(G,\underline W)$ where $codir(Acu_k)$ denotes the category of codirected diagrams in $Acu_k$ as described above.

$$M:\begin{cases}
Acu_k&\to& W(k)-Mod
\\
G&\mapsto&M(G)
\end{cases}$$

is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules.

This construction commutes with automorphisms of $k$. In particular it commutes with the morphism $f_k:k\to k$.
=--

+-- {: .num_defn}
###### Definition and Remark
1. For a $W(k)$-module $M$, define $M^{(p)}:=M\otimes_{W(k),\sigma}W(k)$.

1. As a group $M^{(p)}=M$, but the external law is $(w,m)\mapsto w^{(p^{-1})}m$.

1. If $f\in Acu_k(G,W_{nk})$, then $f^{(p)}:G^{(p)}\to W^{(p)}_{nk}=W_{nk}$ is a morphism and hence a map $\begin{cases}M(G)\to M(G^{(p)})\\f\mapsto f^{(p)}\end{cases}$

1. (...) There is an isomorphism $M(G)^{(p)}\stackrel{\sim}{\to}M(G)^{(p)}$.

1. The Frobenius morphism and the Verschiebung morphisminduce morphisms of $W(k)$ modules. $F:=M(F_G):M(G)^{(p)}\to M(G)$ and $V:=M(V_G):M(G)\to M(G)^{(p)}$.

1. The translation morphism $T:W_{nk}\to W_{(n+1)k}$ is a monomorphism and the maps $Acu_k(G,W_{nk})\to Acu_k(G,W_{(n+1)k})$ are injective.

1. $Acu_k(G,W_{nk})$ can be identified with a submodule of $M(G)$, namely $Acu_k(G,W_{nk})=\{m\in M(G),V^n m=0\}$ and we say that an element of $M(G)$ is killed by a power of $V$.
=--

+-- {: .num_defn}
###### Definition and Remark
Let $D_k$ be the (non-commutative) ring generated by $W(k)$ and two elements $F$ and $V$ subject to the relations

$$Fw=w^{(p)} F$$
$$w^{(p)}V=V w$$
$$FV=VF=p$$

Any element of $D_k$ can be written uniquely as a finite sum

$$\Sigma_{i\gt 0}a_{-i}V^i + a_0 + \Sigma_{i\gt 0}a_i F^i$$

If $G\in Acu_k$, then $M(G)$ has a canonical structure of a left $D_k$-module. If $K$ is a perfect extension of $k$, then there is a canonical map of $D_k$-modules

$$W(K)\otimes_{W(k)}M(G)\to M(G \otimes_k K)$$

Note that $D_K\simeq W(K)\otimes_{W(k)} D_k$ and the left hand side can also be written $D_K\otimes_{D_k}M(G)$.
=--

+-- {: .num_theorem}
###### Theorem
The functor $M$ induces an (contravariant) equivalence

$$Acu_k\to Tor_V D_k Mod$$

between $Acu_k$ and the category of all $D_k$-modules of $V$-torsion.

For any perfect extension $K$ \of $k$ we have that

$$W(K)\otimes_{W(k)}M(G)\to M(G \otimes_k K)$$

is an isomorphism. Moreover

1. $G$ is algebraic iff $M(G)$ is a finitely generated $D_k$-module.

1. $G$ is finite iff $M(G)$ is a $W(k)$-module of finite length.
=--


Demazure, lectures on p-divisible groups, III.6, Dieudonné modules (p-torsion finite k-groups)

[[!redirects III.6, Dieudonné modules (p-torsion finite k-groups)]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Recall from [[relation of certain classes of k-groups]] the following:

1. $Feu_k$ denotes the category of formal étale unipotent affine $k$-groups.

1. $Fiu_k$ denotes the category of formal infinitesimal unipotent $k$-groups.

1. $W(k)$ denotes the Witt ring over $k$.

1. For $D_k$ see [[D_k-module]] in [[III.5, Dieudonné modules (affine unipotent groups)
]].

1. $$M:\begin{cases}
Acu_k&\to& W(k)-Mod
\\
G&\mapsto&M(G)
\end{cases}$$

is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules.

Recall moreover from [[III.5, Dieudonné modules (affine unipotent groups)
]] that $Tor_V D_k-Mod:=Acu_k(G,W_{nk})=\{m\in M(G)|V^n m =0\}$ is a submodule.

which are $W(k)$-modules of finite length, killed by a power of $V$, Definition [[Verschiebung morphism]], and on which $F$, Definition [[Frobenius morphism]], is bijective  (resp. and killed by a power of $F$).

+-- {: .num_prop}
###### Proposition
(formulation of the statement is unclear)
The functor

$$M:\begin{cases}
Acu_k&\to& W(k)-Mod
\\
G&\mapsto&M(G)
\end{cases}$$

which is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules induces the following contravariant equivalences of categories:

1. $Feu_k\to Tor_V D_k-Mod\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $V$ on which $F$ is bijective.

1. $Fiu_k\to Tor_F D_k-Mod\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $F$ (and killed by a power of $V$ ?)on which $F$ is bijective.

([Demazure p.69](#lecturesOnPDivGroups))
=--

+-- {: .proof}
###### Proof
This follows from the theorem, and the fact that if $G$ is finite, then G is étale (resp, infinitesimal) if and only if $F_G$ is an isomorphism (resp. $F_G^n = 0$ for large $n$).
=--

+-- {: .num_example}
###### Example
1. If $G=(\mathbb{Z}/p\mathbb{Z})_k\in Feu_k$, then $M(G)=k$ with $F=1$, $V=0$.

1. If $G=p \alpha_k\in Fiu_k$, then $M(G)=k$ with $F=0$, $V=0$.
=--

+-- {: .num_cor}
###### Corollary
For $G\in Feu_k$ or $G\in Fiu_k$, we have

$$rk(G)= p^{length(M(G))$$
=--

+-- {: .num_prop}
###### Proposition
Let $m,n$ be two positive integers. Then

1. The canonical injection $m^{W_n}\to W_n$ defines an element $u\in M(m^{W_n})$ satisfying $V^n u= F^n= 0$. This gives a map

$$\lambda_{n,m}: D_k/(D_k F^m + D V^n)\to M(m^W_n)$$

which is bijective.
=--

+-- {: .num_theorem}
###### Theorem
There is an isomorphism

$$M(D(G))\to M(G)^*$$

In prose this means that the autoduality $G\mapsto D(G)$ of $Fiu_k$ corresponds via the Dieudonné-functor $M$ to the autoduality $M\to M^*$ in the category $fin Tor_{V,F}D_k-Mod$ of $D_k$-modules of finite length killed by a power of $V$ and $F$.
=--

+-- {: .num_defn}
###### Definition
(Dieudonné-module of an infinitesimal multiplicative $k$-group)

Let $G\in Fim_k$. Then the *Dieudonné-module of $M(G)*$ is defined by

$$M(G)=M(D(G))^*$$

It follows by the [[Cartier duality]] between $Fim_k$ and $Feu_k$ that the functor $G\mapsto M(G)$ induces a contravariant equivalence

$$Fim_k\to  fin Tor_F Bij_V D_k-Mod$$

between $Fim_k$ and the category of all $D_k$-modules of finite length on which $F$ is nilpotent and $V$ is bijective.
=--

+-- {: .num_remark}
###### Remark
Let $G\in Fimd_k$ (i.e. $G\in Fim_k$ and $G$ diagonalizable).

$G=D(\Gamma_k)$. Then $D(G)\simeq \Gamma_k$, and

$M(D(G))=colim Acu_k(\Gamma_k,W_{nk})=colimGr(\Gamma,W_n(k))=Gr(\Gamma,W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z},W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z} \Gamma, W_\infty)$

and hence

$$M(G)\simeq(W(\overline k)\otimes_\mathbb{Z}\Gamma)^\pi$$

where $W_\infty=Quot(W(k))/W(k)=colim_n W_n(k)=\underline W(k)$, see p.66.

For $F$ and $V$ we have

$$F(\lambda \otimes \chi)=\lambda^{(p)}\otimes p\chi$$

$$V(\lambda \otimes \chi)=\lambda^{(p^{-1})}\otimes p\chi$$
=--

+-- {: .num_theorem}
###### Theorem
a) The Dieudonné functor

$$\begin{cases}
F_p_k=Fiu_k\times Feu_k\times Fim_k\to (fin W(k)-Mod,F,V)
\\
G\to M(G)
\end{cases}$$

is a contravariant equivalence between all finite $k$-groups of $p$-torsion, and the category of all triples $(M,F_M,V_M)$ where $M$ is a finite length $W(k)$-module and $F_M$, $V_M$ two group endomorphisms of $M$ satisfying

$$F_M(\lambda m)=\lambda^{(p)} F_M(m)$$

$$V_M(\lambda^{(p)}m)=\lambda V_M(m)$$

$$F_M V_M=V_M F_M=p\cdot id_M$$

b) $G$ is étale, infinitesimal, unipotent or multiplicative according as $F_M$ is isomorphic, $F_M$ is nilpotent, $V_M$ is nilpotent, or $V_M$ is isomorphic

c) For any $G\in Fp_k$ we have $rk(G)=p^{length M(G)}$.

d) If $k$ is a perfect extension of $k$, there exists a functorial isomorphism

$$M(D(G))=M(G)^*$$
=--

## References

{#lecturesOnPDivGroups} Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)


Demazure, lectures on p-divisible groups, III.8, Dieudonné modules (p-divisible groups)

[[!redirects III.8, Dieudonné modules (p-divisible groups)]]
[[!redirects III.8, Dieudonne modules (p-divlsible groups)]]
[[!redirects Dieudonne modules (p-divlsible groups)]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_lemma}
###### Lemma

Let $M_{n+1}\stackrel{\pi_n}{\to}M_n\stackrel{\pi_{n-1}}{\to}\cdots\stackrel{\pi_n}{\to}M_1$ a system of $W(k)$[[W(k)-module|-modules]] such that for all $n$

1. $M_{n+1}\stackrel{p^n}{\to}M_{n+1}\stackrel{\pi_n}{\to}M_n\to 0$ is exact

1. $M_n$ is of finite length.

then $M:=lim M_n$ is a finitely generated $W(k)$-module and the canonical map $M\to M_n$ identifies $M_n\simeq M/p^n M$
=--

+-- {: .num_defn}
###### Definition
($p$-torsion formal group)
A formal group $G$ is called *$p$-torsion formal group* if

1. $G=\cup ker p^n id_G$

1. $ker p id_G$ is finite.

There are exact sequences

$$0\to ker p^n \to ker p^{n+1}\stackrel{p^n}{\to}ker p^{n+1}$$

$$0\to ker p^n \to ker p^{n+m}\stackrel{p^n}{\to}ker p^m$$

showing by induction the also $ker p^n$ is finite for all $n$. Define $M(G)= colim M(ker p^n)$
=--

+-- {: .num_theorem}
###### Theorem

$G\to M(G)$ is a (contravariant) equivalence between the category of $p$[[p-torsion formal group|-torsion]][[formal group|formal groups]] and the category of tuples $(M,F_M,V_M)$ where M
is a finitely generated $W(k)$[[Dieudonné module|-module]] and $F_M$, $V_M$ to groups of endomorphisms of $M$ with

$$F_M(wm)=w^{(p)}F_M (m)$$

$$V_M(w^{(p)}_m)=w V_n(m)$$

$$F_M V_M=V_M F_M=p id_M$$

 It follows from the lemma that $M(G)$ is finitely generated and that

$$M_n=M(G)/p^n M(G)$$

Conversely if $M$ is as before we define $G:=colim G_n$ where $M(G_n)=M/p^n M$

Moreover we have:

1. $G$ is finite iff $M(G)$ is finite and in that case $M(G)$ is the same as in § 7.

1. $G$ is $p$-divisible iff $M(G)$ is torsion-less (= free) and $height(G)=dim M(G)$.

1. For any [[perfect field extension|perfect extension]] $K/k$ there is a functorial isomorphism $M(G\otimes_k K)\simeq W(k)\otimes_{W(k)}M(G)$

1. If $G$ is $p$-divisible with [[Serre dual]] $G^\prime$ then $M(G^\prime)=Mod_{W(k)}(M(G),W(k)$ with

$$(F_{M(G^\prime)}f)(m)=f(V_M m)(p)$$

and

$$(V_{M(G^\prime)}f)(m)=f(F_M m)^{(p^{-1})}$$

([Demazure Theorem p.71-72](#lecturesOnPDivGroups))
=--

+-- {: .num_theorem}
###### Theorem
a) The Dieudonné functor

$$\begin{cases}
Torf_p\to (fin W(k)-Mod,F,V)
\\
G\to M(G)
\end{cases}$$

is a contravariant equivalence between the category of $p$-torsion formal groups, and the category of all triples $(M,F_M,V_M)$ where $M$ is a finitely generated $W(k)$-module and $F_M$, $V_M$ two group endomorphisms of $M$ satisfying

$$F_M(\lambda m)=\lambda^{p} F_M(m)$$

$$V_M(\lambda^{(p)}m)=\lambda V_M(m)$$

$$F_M V_M=V_M F_M=p\cdot id_M$$

It follows from the lemma that $M(G)$ is finitely generated and $M_n\simeq M(G)/p^n M(G)$. Conversely if $M$ is as before, then we define $G$ as $colim G_n$ where $M(G_n)= M/ p^n M$.
=--

+-- {: .num_remark}
###### Remark
From the definition and what we already verified follows:

1. $G$ is finite iff $M(G)$ is finite, and in that case $M(G)$ is the same as defined in § 7.

1. $G$ is finite iff $M(G)$ is torsion-less (= free), and $height(G)=dim M(G)$.

1. For any perfect extension $K/k$, there is a functorial isomorphism $M(G\otimes_k K)\simeq W(K)\otimes_{W(k)} M(G)$.

1. If $G$ is $p$-divisible, with Serre dual $G^\prime$, then $M(G^\prime)=Mod_{W(k)}M(G)$, with $F_{M(G^\prime)} f)(m)=f(V_M m)^{(p)}$ and $(V_{M(G^\prime)}f)(m)=f(F_M m)^{(p^{(-1)}}$.

=--

## References

{#lecturesOnPDivGroups} Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)


Demazure, lectures on p-divisible groups, III.9, Dieudonné modules (connected formal groups of finite type)

[[!redirects III.9, Dieudonné modules (connected formal groups of finite type)]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

By a similar discussion (replacing $p$ by $F$) as in § 8, we have:

+-- {: .num_remark}
###### Remark
If $G$ is is a connected finite type formal group, define

$$M(G):= lim M(ker F^n_G)$$

This is a module over the $F$-completion $\hat D_k$ of $D_k$.
=--

+-- {: .num_theorem}
###### Theorem
The Dieudonné-functor is an equivalence

$$\begin{cases}
Fftc\to \hat D_k-Mod_{fin.len.quot}
\\
G\mapsto M(G)
\end{cases}$$

between the category of connected formal groups of finite type and the category of $\hat D_k$-modules $M$ such that $M/FM$ has finite length. Moreover we have:

1. $G$ is finite iff $M(G)$ has finite length iff $F^n M(G)=0$ for large $n$.

1. $G$ is smooth iff $F:M(G)\to M(G)$ is injective. In that case $dim(G)= length(M(G)/ FM(G))$.
=--


Demazure, lectures on p-divisible groups, IV.1, isogenies

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Unless otherwise stated let $k$ be a perfect field of prime characteristic.

We denote write $B(K):=Quot(W(k))$ for the [[quotient field]] of the [[Witt ring]] $W(k)$. We extend the [[Frobenius morphism]] $x\mapsto x^{(p)}$ to an automorphism of $B(k)$. The set of fixed points of $x\mapsto x^{(p)}$ in $W(k)$ is $W(F_p)=\mathbb{Z}_p$. The set of fixed points of $x\mapsto x^{(p)}$ in $B(k)$ is $B(F_p)=\mathbb{Q}_p$.


Demazure, lectures on p-divisible groups, V.2, points of finite order and endomorphisms

Let $k$ be a field of prime characteristic $p$.

+-- {: .num_defn}
###### Definition

Let $A$ be an [[abelian variety]] of dimension $g$. Let $l$ be a prime number.

For $n\in \mathbb{N}$ the kernel $ker l^n id_A)$ is a finite $k$-group of rank $l^{2 n g}$. We define

$$A(l):=\cup_n ker(l^n id_A)$$

We have $A(l)\otimes_k \overline k\simeq (\mathbb{Q}(\mathbb{Z}_l))^{2g}$. If $l\neq p$, then $A(l)$ is an [[étale formal k-group]].
=--

+-- {: .num_defn}
###### Definition
We define

$$H^1(A,l):=hom_{\mathbb{Z}_l}(A(l)\otimes_k \overline k, \mathbb{Q}_l /\mathbb{Z}_l)$$

This is a free module of rank $2g$ over $\mathbb{Z}_l$ and also a [[Galois module]].

If $l=p$, then $A(p)$ is a $p$-divisible group of height $2g$. In this case we define

$$H^1(A,p):=M(A(p))$$

as the [[Dieudonné module]] of $A(p)$. It is an [[F-lattice]] (defined in [[Demazure, lectures on p-divisible groups, IV.1, isogenies]]) over $k$, and in particular a free module of rank $2g$ over $W(k)$.

For any prime $l$, the assignation

$$A\mapsto H^1(A,l)$$

is a functor.
=--


Demeter Krupka

* [webpage](http://www.lepageri.eu/research-group/demeter-krupka)

## Selected writings

On [[jet bundles]] and [[natural bundles]]/[[natural operations]] in [[differential geometry]]:

* [[Demeter Krupka]], [[Josef Janyška]], Part 2 of: _Lectures on differential invariants_, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, [researchgate](https://www.researchgate.net/publication/36792711_Lectures_on_Differential_Invariants)]

## Related entries

* [[differential geometry]]

* [[jet group]], [[higher order frame bundle]]

* [[variational sequence]], [[Lepage form]]

* [[jet group]]

category: people


Democritus

* [Wikipedia entry](https://en.wikipedia.org/wiki/Democritus)

## Selected claims

On the nature of the [[Milky Way]]:

>  Democritus, $[$holds$]$ that it is the splendor which ariseth from the coalition of many small *asterons*, which, being firmly united amongst themselves, do mutually enlighten one another.

See the original Greek [here](http://articles.adsabs.harvard.edu/pdf/1901Obs....24..382L#page=2), from:

* {#PseudoPlutarch} [Pseudo-Plutarch](https://en.wikipedia.org/wiki/Pseudo-Plutarch), *[Of the Galaxy, or the Milky Way](http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0404%3Abook%3D3%3Achapter%3D1)*, Book 3, Chapter I in:  *[Placita Philosophorum](http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2008.01.0404)*, ~300-400 AD

category: people


demonstration of the prime ideal theorem

let L be a prime ideal, our aim is to show that there exist


dendrex

#Contents#
* table of contents
{:toc}

## Idea

The notion of _dendrex_ is the [[operad|operadic]] analog of the notion of [[simplex]].

## Definition

For $X$ a [[dendroidal set]] and $T$ a [[tree]], a **$T$-shaped dendrex** of $X$ is a morphism

$$
  \Omega[T] \to X
  \,,
$$

hence equivalently, by the [[Yoneda lemma]], an element of the set $X(T)$.

[[!redirects dendrices]]


dendriform algebra

Dendriform algebra is a generic term both for dendriform dialgebras and dendriform trialgebras.

__Dendriform dialgebra__s are algebras over a Koszul quadratic operad which is Koszul dual to the [[associative dialgebra]] operad. They have two binary operations $\prec$, $\succ$ (commonly also denoted $\dashv$ and $\vdash$) satisfying 3 quadratic relations. The sum of the two binary operations is an associative operation.

Dendriform dialgebras were introduced by Loday in his study of K-theory,

* [[Jean-Louis Loday]], _Dialgebras_, in: Dialgebras and Related Operads, Springer Lecture Notes in Mathematics 1763 (2001) 766, 1--61

* [[Kurusch Ebrahimi-Fard]], [[Li Guo]], _Rota--Baxter algebras and dendriform algebras_, J. Pure Appl. Alg. __212__:2 (2008) 320-339
* Sara Maradiaga, _Splitting of operations for alternative and Malcev structures_, [arXiv:/1312.5710](https://arxiv.org/abs/1312.5710)

category: algebra

[[!redirects dendriform algebras]]
[[!redirects dendriform dialgebra]]
[[!redirects dendriform trialgebra]]


dendroidal homotopy coherent nerve

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _dendroidal homotopy coherent nerve_ is an [[operad|operadic]] generalization of the standard [[homotopy coherent nerve]]. It is a [[functor]]

$$
  hcN_d : Top Operad \to dSet
$$

from the [[category]] of [[Top]]-[[operads]] to that of [[dendroidal sets]], given by

$$
  hcsN_d(P) : T \mapsto dSet(W_H(T), P)
  \,,
$$

where $T$ is an object of the [[tree category]], regarded as a free [[symmetric operad]], and $W_H(T)$ is its [[Boardman-Vogt resolution]].

## Definition

Throughout, let $\mathcal{E}$ be a [[symmetric monoidal category|symmetric]] [[monoidal model category]] equipped with an [[interval object]] $H$ as discussed at _[[model structure on operads]]_ and  at _[[Boardman-Vogt resolution]]_. We consider multi-coloured [[symmetric operads]] ([[symmetric multicategories]]) enriched in $\mathcal{E}$.

Standard examples are $\mathcal{E} = $ [[Top]], [[sSet]], which yields [[topological operads]] and [[simplicial operads]], respectively.

### BV resolution of trees
 {#BVResolutionOfTrees}

We discuss in detail what the [[Boardman-Vogt resolution]] of [[operads]] free on an object in the [[tree category]] $\Omega$ is like (see _[[dendroidal set]]_ for details on trees as operads).

Let

$$
  Symm : \mathcal{E} Operad_{planar} \to \mathcal{E} Operad
$$

be the symmetrization functor, the [[left adjoint]] to the [[forgetful functor]] from [[symmetric operads]] to [[planar operads]].

+-- {: .num_prop}
###### Proposition

The BV resolution commutes with symmetrization: if $T = Symm(\bar T)$, then

$$
  W(T) = Symm(W(\bar T))
  \,.
$$

=--

Therefore we describe in the following explicitly the BV-resolution of planar trees, that of non-planar trees then being the symmetrization of that construction.

+-- {: .num_prop #ComponentDescriptionofBVofTrees}
###### Proposition

For $T \in \Omega_{planar}$, and $(e_1, \cdots, e_n; e)$ a tuple of colours (edges) of $T$, notice that the set of operations $T(e_1, \cdots, e_n, e)$ is the set of those subtrees $V \subset T$ such that $\{e_1, \cdots, e_n\}$ is the set of leaves and $e$ is the root of $V$.

First regard $T$ as a [[topological operad]] (with a [[discrete space]] of operations in each degree). The corresponding [[Boardman-Vogt resolution]] $W(T)$ of $T$ is the topological operad whose [[topological space]] of operations $W(T)(e_1, \cdots, e_n; e)$ is the space of _labeled trees_ as follows.

A point is a set of _lengths_ $\ell(e) \in [0,1]$, one for each inner edge $e \in I(T)$ of $T$. (...)

Hence

$$
  W(T)(e_1, \cdots, e_n; e)
    \simeq
  \coprod_{V \in T(e_1, \cdots, e_n; e)}
  (\Delta^1)^{\times i(V)}
$$

where the [[coproduct]] ranges over the set of subtrees $V$, as just discussed (which therefore is either the singleton set or is empty), and where $i(V)$ is the set of inner edges of $V$.

Regard then $T$ as a [[simplicial operad]]. The corresponding [[Boardman-Vogt resolution]] $W(T)$ of $T$ is the simplicial operad whose [[simplicial sets]] of operations are

$$
  W(T)(e_1, \cdots, e_n; e)
  =
  \coprod_{V \in T(e_1, \cdots, e_n; e)}
  \Delta[1]^{\times i(V)}
  \,.
$$

In general, when $T$ is regarded as an $\mathcal{E}$-operad, we have

$$
  W(T)(e_1, \cdots, e_n; e)
  =
  \coprod_{V \in T(e_1, \cdots, e_n; e)}
  H^{\otimes i(V)}
  \,,
$$

where $H$ is the given [[interval object]].

=--

+-- {: .num_prop #CompositioninWT}
###### Observation

The composition operations in $W(T)$

$$
  \array{
    W(T)(e_1, \cdots, e_n; e_0) \otimes W(T)(f_1, \cdots, f_k; e_i)
    \\
    \downarrow^{\mathrlap{\circ_i}}
    \\
    W(T)(e_1, \cdots, e_{i-1}, f_1, \cdots, f_k, e_{i+1}, \cdots, e_n)
  }
$$

correspond to grafting of trees $T_\sigma, T_\rho \subset T$ and "assigning unit length to the new inner edge". On the components as discussed above it is given by

$$
  \array{
    H^{\otimes i(T_\sigma)} \otimes H^{\otimes i(T_\rho)}
    &&
    H^{\otimes i(T_\sigma \circ_i T_\rho)}
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    H^{\otimes i(T_\sigma) \cup i(T_\rho)} \otimes I
     &\stackrel{id \otimes i_1}{\to}&
    H^{\otimes (i(T_\sigma) \cup i(T_\rho))} \otimes H
  }
$$

=--

+-- {: .num_prop #BVResolutionAsFunctorOnOmega}
###### Proposition

The BV-resolution of trees extends to a [[functor]] on the category of [[simplicial operad]]

$$
  W : \Omega \to sSet Operad
$$

as follows

* on an inner face map $\delta_e : \partial^e\Omega[T] \to \Omega[T]$ the component of $W(\delta)$ on a subtree $V$ of $T$ that contains the edge $e$ is the product of the inclusion

  $$
    i_0 : I \to H
  $$

  with the identity on $H^{i(V)-\{e\}}$

  (meaning: if the label of an inner edge in a tree is 0, then the operations that it connects may be composed);

* on a degenracy map $\sigma $ that sends two given unary vertices to a single one, the component of $W(\sigma)$ on subtrees containing these removes one of the factors $H$ by the map

  $$
    H \otimes H \to H
  $$

  given by the [[interval object]] $H$. For both [[simplicial operads]] and [[topological operads]] this may be taken to be the map

  $$
    max : \Delta^1 \times \Delta^1 \to \Delta^1
  $$

  that sends $(x,y)$ to $max(x,y)$.

=--

This is discussed in section 4.2 of ([Cisinski-Moerdijk](#CisinskiMoerdijk)).

### The homotopy coherent nerve

By the general discussion at _[[nerve and realization]]_, the functor

$$
  W
  :
  \Omega
  \to
  sSet Operad
$$

from prop. \ref{BVResolutionAsFunctorOnOmega} induces a [[nerve]] functor as follows.

+-- {: .num_defn}
###### Definition

The **dendroidal homotopy coherent nerve** functor is the [[functor]]

$$
  hcN_d : sSet Cat \to dSet
$$

given by

$$
  P \mapsto ( T \mapsto sSet Operad(W(T), P) )
  \,.
$$

Its [[left adjoint]] (the corresponding "[[geometric realization]]") we denote

$$
  W_! : dSet \to sSet Operad
  \,.
$$

=--

## Properties

### Specialization to categories

+-- {: .num_prop}
###### Proposition

When restricted to $\mathcal{E}$-[[enriched categories]], the dendroidal homotopy coherent nerve reproduces the [[homotopy coherent nerve]] of enriched categories

$$
  \array{
    \mathcal{E} Cat &\hookrightarrow& \mathcal{E} Operad
    \\
    {}^{\mathllap{hcN}}\downarrow && \downarrow^{\mathrlap{hcN_d}}
    \\
    sSet &\hookrightarrow& dSet
  }
  \,.
$$

In particular for $\mathcal{E} = $ [[Top]] / [[sSet]] it reproduces the original definition of homotopy coherent nerve.

=--

### Dendroidal inner Kan complexes
 {#DendroidalInnerKanComplexes}

+-- {: .num_prop #hcNOfLocllyFibrantOperadIsFibrant}
###### Proposition

Let $P \in \mathcal{E} Operad$ be such that each object of operations is fibrant in $\mathcal{E}$. Then its homotopy coherent nerve $hcN_d(P)$ is a [[model structure on dendroidal sets|dendroidal inner Kan complex]].

=--

This is ([Moerdijk-Weiss, theorem 7.1](#MoerdijkWeiss)).
This statement will also follow as a corollary from prop. \ref{WIsLeftQuillen} below.

+-- {: .proof}
###### Proof

Consider a tree $T$ and an inner edge $e$ of it. For each morphism
$\phi : \Lambda^e[T] \to X$ we need to find a filler $\psi$ in

$$
  \array{
    \Lambda^e[T] &\to& hcN_d(X)
    \\
    \downarrow & \nearrow_{\mathrlap{\psi}}
    \\
    \Omega[T]
  }
  \,.
$$

Write $\Lambda^e[T] = \cup_{i \neq e}\partial^{i \neq e} \Omega[T]$.

By the definition of dendroidal nerve, this is equivalently a diagram

$$
  \array{
    \cup_{i \neq e } W(\partial^i \Omega[T]) &\to& X
    \\
    \downarrow & \nearrow_{\mathrlap{\hat \psi}}
    \\
    W(\Omega[T])
  }
  \,.
$$

The undetermined component to fill is that corresponding to the subtree $\tau$ of $T$ which is $T$ itself. According to prop. \ref{ComponentDescriptionofBVofTrees} on this
the operad $W(\Omega[T])$ has the component

$$
  H^{\otimes i(\tau)} \simeq H^{\otimes i(\tau)- \{e\}}\otimes H
  \,.
$$

The map $\hat \psi$ has to send this into $X$ while being compatible with the given faces. By prop. \ref{BVResolutionAsFunctorOnOmega} this means that its precomposition with all the inclusions

$$
  (id, \cdots, id, i_0, id, \cdots, id) \otimes id
  :
  H \otimes \cdots \otimes H \otimes I \otimes H \otimes \cdots \otimes H \otimes H
  \to
  H^{\otimes i(\tau)- \{e\}}\otimes H
$$

is fixed. Moreover, the assignment needs to be compatible with the composition operations, which by prop. \ref{CompositioninWT} means that also the precomposition with all the maps

$$
  (id, \cdots, id, i_1, id, \cdots, id)
  :
  H \otimes \cdots \otimes H \otimes I \otimes H \otimes \cdots \otimes H
  \to
  H^{\otimes i(\tau)}
$$

is fixed. In total this means that the components of $\hat \psi$ need to form an extension of the form

$$
  \array{
    (\partial H^{\otimes i(\tau)- \{e\}})
    \otimes H
    \cup
    H^{\otimes i(\tau) - \{e\}}\otimes I
    &\to& X(\tau)
    \\
    \downarrow & \nearrow
    \\
    H^{\otimes i(\tau)}
  }
$$

in $\mathcal{E}$, where

$$
  \partial H^n :=
    (I \coprod I) \otimes H^{n-1}
    \coprod
    H (I \coprod I) \otimes H^{n-2}
    \coprod
    \cdots
    \stackrel{(i_0,i_1)\otimes id \coprod \cdots}{\to}
    H^n
  \,.
$$

One sees that the left vertical morphism is an acyclic cofibration, by the [[pushout-product axiom]] in the [[monoidal model category]] $\mathcal{E}$. Therefore by the assumption that $X(\tau)$ is fibrant, such a lift does exist.

=--

### Left adjoint
  {#LeftAdjoint}

+-- {: .num_defn}
###### Definition

Write

$$
  W_! : dSet \to sSet Operad
$$

for the [[Yoneda extension]] of

$$
  \Omega \hookrightarrow Operad \hookrightarrow sSet Operad \stackrel{W}{\to} sSet Operad
  \,;
$$

hence for the [[functor]] from [[dendroidal sets]] to [[simplicial operads]], which

* preserves [[colimits]];

* on trees $\Omega \hookrightarrow Operad \hookrightarrow sSet Operad$
  is given by the [[Boardman-Vogt resolution]] as discussed [above](#BVResolutionOfTrees).

=--

By the general lore of [[nerve and realization]] we have

+-- {: .num_prop}
###### Proposition

$W_!$ is [[left adjoint]] to $hcN_d$

$$
  (W_! \dashv hcN_d)
  :
  sSet Operad
  \stackrel{\overset{N_d}{\leftarrow}}{\underset{hcN_d}{\to}}
 dSet
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

For $P \in sSet Operad$, the [[unit of an adjunction|counit]]

$$
  W_! hcN_d (P) \to P
$$

is essentially the [[Boardman-Vogt resolution]] of $P$.

For a cofibrant and fibrant $X \in dSet$, the [[unit of an adjunction|unit]]

$$
  X \to hcN_d W_!(X)
$$

may be viewed as a "strictification" of the [[(infinity,1)-operad]] given by $X$, in that  $W_!(X)$, being a simplicial operad, has strictly associative composition.

=--

### Relation to ordinary dendroidal nerve

By the general properties of the [[Boardman-Vogt resolution]]
(but also immediately checked directly) we have

+-- {: .num_prop #CounitOfResolutionOnTrees}
###### Proposition

There is a [[natural transformation]]

$$
  \epsilon : W \Rightarrow \Omega(-) : \Omega \to sSet Operad
$$

$$
  \epsilon_T : W(T) \to T
$$

(natural in the tree $T \in \Omega$), which is a bijection on colors and is
on the components of prop. \ref{ComponentDescriptionofBVofTrees} the canonical map

$$
  \Delta[1]^{i(V)} \to *
  \,.
$$

Each $\epsilon_T$ is hence a weak equivalence of simplicial operads. In particular

$$
  \pi_0(W(T)) \to T
$$

is an [[isomorphism]].

=--

This induces hence a [[natural transformation]]

$$
  W_! \Rightarrow \tau_d : dSet \to sSet Operad
$$

to the left adjoint $\tau_d$ of the ordinary [[dendroidal nerve]] (the "fundamental operad" construction).

+-- {: .num_prop}
###### Proposition

For every [[dendroidal set]] $X$, the natural morphism

$$
  \pi_0 W_!(X) \to \pi_0 \tau_d (X) = \tau_d(X)
$$

is an [[isomorphism]] of simplicial operads.

=--

This appears as [Cisinski-Moerdijk, prop. 4.4](#CisinskiMoerdijk).

+-- {: .proof}
###### Proof

The functors $\pi_0 W_!$ and $\tau_d$, being [[left adjoints]], both preserve small [[colimits]]. Therefore it is sufficient to check the statement for $X = \Omega[T]$ a tree. There it is prop. \ref{CounitOfResolutionOnTrees}.

=--

### Quillen equivalence

+-- {: .num_theorem }
###### Theorem

The [[adjunction]] $(W_! \dashv hcN_d)$ from [above](#LeftAdjoint) is a [[Quillen equivalence]] between the [[model structure on operads]] over [[Top]]/[[sSet]] and the [[model structure on dendroidal sets]].

=--

We discuss some input to this statement.

+-- {: .num_prop #WIsLeftQuillen}
###### Proposition

The functor $W_! : dSet \to sSet Operad$ sends normal monomorphisms to cofibrations, and inner
[[anodyne extensions]] to [[acyclic cofibrations]] in the [[model structure on sSet-operads]].

=--

This appears as [Cisinski-Moerdijk, prop. 4.5](#CisinskiMoerdijk).

+-- {: .proof}
###### Proof

Observe that the morphism classes in question are, as discussed at _[[dendroidal set]]_, the saturated classes generated by the dendroidal boundary inclusions and by the dendroidal horn inclusions, respectively.

Since $W_!$ is left adjoint, it therefore suffices to check the statement on these generating inclusions. Moreover, by construction, on trees $W_!$ coincides with the [[Boardman-Vogt resolution]] of the operads free on these trees.

It follows that the generating inclusions are sent by $W_!$ to morphisms of simplicial operads which are

* bijective on objects;

* isomorphisms on all but one simplicial set of operations: that corresponding to the maximal subtree;

* on this remaining simplicial set of operations a product of identities with cofibrations of simplicial sets (monomorphisms), and following through the combinatorics shows that these are acyclic for the case of anodyne extensions.

It follows that these morphisms of simplicial operads have the left lifting property again operation-object-wise [[Kan fibrations]] (there is no further composition to be respected, since the maximal subtree operation has no further non-trivial composites), and hence against the fibrations of the [[model structure on sSet-operads]].

=--

+-- {: .num_remark}
###### Remark

Prop.\ref{hcNOfLocllyFibrantOperadIsFibrant} is, in turn, a direct consequence of this.

=--

### Resolution and rectification

+-- {: .num_prop}
###### Proposition

Let $P$ be an [[operad]] in [[Set]], regarded as an $\mathcal{E}$-operad. Then the $(W_! \dashv hcN_d)$-[[unit of an adjunction|counit]]

$$
  W_! N_d(P) = W_! hcN_d (P) \to P
$$

is isomorphic to the [[Boardman-Vogt resolution]] $W_H(P)$ of $P$.

In particular, therefore, there is a [[natural isomorphism]]

$$
  Hom_{\mathcal{E}Operad}(W_H(P), Q)
  \simeq
  Hom_{dSet}(N_d(P), hcN_d(Q))
  \,.
$$

=--

(Here we are using that on a discrete operad $P$ the homotopy coherent dendroidal nerve trivially coincides with the ordinary dendroidal nerve $N_d$.)

+-- {: .proof}
###### Proof

By inspection of the relevant formulas.

=--

+-- {: .num_prop}
###### Proposition

For a cofibrant and fibrant dendroidal set $X$, the $(W_! \dashv hcN_d)$-[[unit of an adjunction|unit]]

$$
  X \to hcN_d W_! X
$$

is an equivalence.

=--

+-- {: .num_remark}
###### Remark

Since composition of operations in a simplicial operad is strictly associative,  this may be understood as producing a semi-strictification of the $\infty$-operad $X$.

=--

## Related concepts

[[!include table - models for (infinity,1)-operads]]

## References

The fact that the homotopy coherent nerve if a locally fibrant operad is inner Kan is shown in section 7 of

* [[Ieke Moerdijk]], [[Ittay Weiss]], _On inner Kan complexes in the category of dendroidal sets_, ([math.CT/0701295](http://arxiv.org/abs/math/0701295))
 {#MoerdijkWeiss}

The Quillen adjunction properties of the homotopy coherent dendroidal nerve are discussed in section 4 of

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets and simplicial operads_ ([arXiv:1109.1004](http://arxiv.org/abs/1109.1004))
 {#CisinskiMoerdijk}

Lecture notes on these two topics are in section 6 and 9 of

* [[Ieke Moerdijk]], _Lectures on dendroidal sets_ , lectures given at the Barcelona workshop on _[Simplicial methods in higher categories](http://www.crm.es/HigherCategories/)_ (2008) ([preliminary writeup](http://www.crm.cat/HigherCategories/hc1.pdf))

[[!redirects homotopy coherent dendroidal nerve]]


dendroidal set

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

Dendroidal sets are a [[geometric definition of higher categories|geometric]] model for [[(∞,1)-operad|higher]] _[[operads]]_ (precisely: multi-coloured [[symmetric operads]] / [[symmetric multicategories]]).
They are to [[operads]] and to [[(∞,1)-operads]] as [[simplicial sets]] are to [[category|categories]] and [[(∞,1)-categories]].

A dendroidal set is something that consists of [[tree category|trees]] or "[[dendrices]]" in the way that a [[simplicial set]] consists of [[simplex|simplices]]: the trees represent the [[free construction|free]] [[Set]]-[[operads]] over them, and so a dendroidal set is a structure defined as having consistent probes by free $Set$-operads.

More precisely, the [[category]] [[dSet]] of dendroidal sets serves to complete the following [[commuting diagram]] of [[functors]]

$$
  \array{
    Cat &\hookrightarrow& Operad
    \\
    {}^{\mathllap{N}}\downarrow &&
    {}^{\mathllap{N_d}}\downarrow
    \\
    sSet
    &\hookrightarrow&
    dSet
  }
  \,,
$$

where the left vertical functor is the [[nerve]] $N :$ [[Cat]] $\to$ [[sSet]], and where the top horizontal functor includes categories into [[Set]]-enriched operads as those operads with only unary operations.
Moreover, $dSet$ completes this diagram even as a diagram of most of the evident pairs of [[adjoint functors]]. For details see _[The full diagram of relations](#FullDiagram)_ below.

Finally, there is a [[model structure on dendroidal sets]] which is the operadic analog of the [[model structure for quasi-categories]].

## Definition ##

### Dendroidal sets
 {#DendroidalSets}

+-- {: .num_defn }
###### Definition

Write [[Operad]] for the category of [[operads]], specifically for

* [[Set]]-enriched;

* [[symmetric operad|symmetric]];

* multi-coloured

operads, also known as _[[symmetric multicategories]]_.

=--

+-- {: .num_defn #FreeOperadCategory}
###### Definition

The symmetric [[tree category]] is the [[full subcategory]]

$$
  \Omega \hookrightarrow Operad
$$

of [[Operad]] on those [[symmetric operads]] that are [[free construction|free]] on finite rooted non-planar [[trees]].

=--

See ([below](#Trees)) for details on the trees appearing here.

+-- {: .num_defn #DendroidalSet}
###### Definition

A **dendroidal set** is a [[presheaf]] on the [[tree category]] $\Omega$.

The [[category]] of dendroidal sets is the [[functor category]]

$$
  dSet = [\Omega^{op}, Set]
  \,.
$$

=--

Some **terminology and notation**:

* For $T \in \Omega$ a tree, write $\Omega[T] := \Omega(-,T)$ for the dendroidal set that it [[representable functor|represents]].

* For $X$ a dendroidal set and $T$ a tree, the set $dSet(\Omega[T],X) \simeq X(T)$ (using the [[Yoneda lemma]]) is the set of $T$-shaped **[[dendrices]]** in $X$. One of its elements is called a $T$-shaped **dendrex**, analogous to a [[simplex]] in a [[simplicial set]].

### Trees and their free operads
 {#Trees}

We expand on the notion of symmetric/non-planar trees used in definition \ref{DendroidalSet}
of dendroidal sets (see for instance ([Weiss, section 2.1](#Weiss))).

+-- {: .num_defn }
###### Definition

A **finite symmetric rooted tree** -- or just **tree** for short, in the following --  is a [[finite set|finite]] [[poset]] $(T, \leq)$
such that

* it has a [[bottom]] element;

* for each element $e \in T$ the set $\{y \in T | y \leq e\}$ is a [[linear order]] under $\leq$;

and equipped with a subset $L \subset max(T)$ of the maximal elements of $T$.

=--

One has the following terminology

+-- {: .num_defn }
###### Terminology

Let $((T, \leq), L)$ be a tree.

* An element $e \in T$ is called an **edge** of the tree.

* The [[bottom]] element is called the **root** of the tree.

* An edge in $L \subset T$ is a **leaf** of the tree.

* An edge which is either a leaf or the root is called an **outer edge**.

* An edge which is neither a leaf nor the root is called an **inner edge**.

* Given a non-leaf edge $e$, another edge $e'$ is called an **incoming edge** of $e$ if it is a direct succesor of $e$, hence if

  1. $e \leq e'$;

  1. for any $e \in T$ with $e \leq x \leq e'$ either $x = e$ or $x = e'$.

  Write $in(e)$ for the set of incoming edges of $e$.

* For $e$ a non-leaf edge, the subset $v_e := \{e\} \cup in(e)$ is called a **vertex** of $(T, \leq)$. $e$ is called the **outgoing edge** of $v$, and the ingoing edges of $e$ are also called _ingoing edges of $v_e$_, $in(v_e) := in(e)$.

* The **valence** of a vertex $v$ is the [[cardinality]] ${\vert in(v)\vert}$.

=--

+-- {: .num_example #ATypicalTree}
###### Example

A typical tree as above is

$$
  \left\{
     \array{
        {}_{\mathllap{e}}\searrow && \swarrow_{\mathrlap{f}}
        \\
        & \bullet_v
        \\
        && {}_{\mathllap{b}}\searrow && \swarrow_{\mathrlap{c}}
        \\
        & && \bullet_u & \stackrel{d}{\leftarrow} & \bullet_w
        \\
        & && \downarrow^{\mathrlap{r}}
     }
  \right\}
  \,,
$$

where

* the arrows depict the edges;

* the bullets depict vertices;

* the arrows not starting at a bullet depict leaf edges.

* the arrow not ending at a bullet depicts the root edge.

The set of incoming edges of $v$ is $\{e,f\}$, the outgoing edge of $v$ is $b$. The set of incoming edges of $u$ is $\{b,c,d\}$, its outgoing edge is the root edge $r$. The set of ingoing edges of $e$ is the [[empty set]], its outgoing edge is $d$.

The valence of $v$ is 2, that of $u$ is 3 and that of $w$ is 0.

Notice that there is _no_ order on the incoming edges of any vertex implied, the fact that there is one in the above picture is an artefact of being a planar diagram for visualization purposes. As a tree, the above is _equal_ to, for instance

$$
  \left\{
     \array{
        {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{e}}
        \\
        & \bullet_v
        \\
        && {}_{\mathllap{b}}\searrow && \swarrow_{\mathrlap{c}}
        \\
        & && \bullet_u & \stackrel{d}{\leftarrow} & \bullet_w
        \\
        & && \downarrow^{\mathrlap{r}}
     }
  \right\}
  \,,
$$

=--

+-- {: .num_defn }
###### Definition

For $((T, \leq ), L)$ a tree, the corresponding [[symmetric operad]]
over [[Set]] is the one

* whose set of colours is is $T$ (the set of edges);

* whose operations are freely generated from the set of vertices of $T$, where the generating operation corresponding to a vertex $v$ goes from $in(v)$ to the outgoing edge $c$ of $v$.

More precisely, for every vertex $v$ and every choice of order
$(c_1, \cdots, c_n)$ on $in(v)$ there is a generating operation

$$
  (c_1, \cdots, c_n) \to c
$$

and the action of an element of the [[symmetric group]]
$\sigma \in S_n$ takes this to the generator

$$
  (c_{\sigma(1)}, \cdots, c_{\sigma(n)}) \to c
  \,.
$$

=--

+-- {: .num_example }
###### Example

The operad corresponding to the tree from example \ref{ATypicalTree}

* has six colours;

* has precisely six non-identity operations:

  * the generators $v$, $u$, $w$;

  * the composites $u \circ (v, id_c, id_d)$ and $u \circ (id_b, id_c, w)$ and $u \circ (v, id_c, w)$.

=--

The inclusion $\Omega \hookrightarrow$ [[Operad]] is the [[full subcategory]] on those operads that arise from trees in this way.

Some non-typical trees of importance are the following.

+-- {: .num_example #LinearTree}
###### Example

For every $n \in \mathbb{N}$ there is the **linear tree**

$$
  L_n
  :=
  \left\{
    \array{
      \downarrow^{0}
      \\
      \bullet_{01}
      \\
      \downarrow^{1}
      \\
      \bullet_{02}
      \\
      \vdots
      \\
      \downarrow^{n}
    }
  \right\}
$$

with $(n+1)$ colors and only unary operations.

The map that sends the [[simplicial set|simplicial]] [[simplex]] $\Delta^n$ to $L_n$
extends to a functor

$$
  \Delta \hookrightarrow \Omega
$$

which exhibits the [[simplex category]] as a [[full subcategory]] of the [[tree category]].

=--

+-- {: .num_example #Corolla}
###### Example

For each $n \in \mathbb{N}$ there is the **corolla tree**

$$
  C_n :=
  \left\{
    \array{
      {}_{\mathrlap{l_1}}\searrow & \cdots & \swarrow_{\mathrlap{l_n}}
      \\
      & \bullet
      \\
      & \downarrow
    }
  \right\}
$$

with $n$ leaves and precisely one vertex.

=--

+-- {: .num_remark }
###### Remark

The trees $L_0$ and $C_0$ differ. $L_0$ is the tree

$$
  \left\{
    \downarrow
  \right\}
$$

with no vertex, while $C_0$ is the tree

$$
  \left\{
    \array{
      \bullet
      \\
      \downarrow
    }
  \right\}
$$

with a single vertex, which has valence 0.

=--

### Operations -- Faces and horns of trees
 {#OperationsFacesHorns}

A dendroidal set encodes composition of _operations_ in analogy to how a [[simplicial set]] encodes composition of edges: by way of _[[horn]] extensions_. In order to formalize this one uses the dendroidal analog of _faces_ of a [[simplex]], generalized from the [[simplex category]] to the [[tree category]].

To motivate the definition of dendroidal face maps, first consider the reformulation of simplicial faces under the map $i_! : sSet \to dSet$.

Under this embedding the $n$-simplex $\Delta[n]$ becomes the operad corresponding to the linear tree $L_n$, example \ref{LinearTree}

$$
  \stackrel{a}{\to} \bullet_{01} \stackrel{b}{\to} \bullet_{02} \stackrel{c}{\to} \cdots \to \bullet_{(n-1) n} \to
$$

with unary operations $\bullet_{i (i+1)}$.

The face inclusion $\delta_i : \Delta[n-1] \to \Delta[n]$ which simplicially omits the $i$th vertex, operadically contracts away the $i$th color, i.e it is the morphism of operads that sends the unary operation $\bullet_{(i-1)i}$ to the unary operation

$$
  \bullet_{i (i+1)} \circ \bullet_{(i-1) i }
$$

and sends all other generating unary operation to generating unary operations.

From this it is clear that for any tree the map that exhibits the contraction of one edge should be a dendroidal face map. However, there are also some more cases to be taken care of. One has

1. inner face maps -- obtained by contracting an inner edge

1. outer face maps -- obtained by

   1. removing an outer input vertex

   1. removing a vertex whose output is the root and which has precisely one inner incoming edge (which becomes the new root)

   1. a corolla face -- any one of the inclusions of the tree with no vertex into a tree with precisely one vertex

+-- {: .num_defn }
###### Definition

For $T \in \Omega$ a tree and $e$ an _inner edge_ , write $T/e$ for the tree obtained by contracting/discarding this inner edge. There is a canonical inclusion

$$
  \partial_e : T/e \to T
$$

in $\Omega$.

This is called an **inner face map** of $T$.

=--

+-- {: .num_example #ExampleForInnerFace}
###### Example

Let

$$
  T =
   \left\{
    \array{
     \searrow && \swarrow
     \\
     & v
     \\
     \searrow & & \searrow^e
     \\
      & \bullet &\to& \bullet &\to&
     \\
     \nearrow
    }
  \right\}
$$

then

$$
  T/e =
   \left\{
    \array{
     \searrow && \searrow& & \swarrow
     \\
      & \bullet &\to& \bullet &\to&
     \\
     \nearrow
    }
  \right\}
$$

and the inclusion $T/e \to T$ is the operad morphism that is the identity on the operation  $
  \array{
    \searrow
    \\
    & \bullet & \to
    \\
    \nearrow
  }
$
and which sends the trinary operation
$
  \array{
     \searrow && \swarrow
     \\
     \to & \bullet & \to
  }
$
to the composite trinary operation
$
    \array{
     \searrow && \swarrow
     \\
     & \bullet
     \\
      & & \searrow^e
     \\
      & &\to& \bullet &\to&
     \\
    }
$

=--

In contrast to that, _outer_ edges are always removed together:

+-- {: .num_defn }
###### Definition

If $v$ is a vertex of $T$ such that all but one edge incident on it are outer, then denote by $T/v$ the tree obtained by discarding $v$ and all the outer edges indicent on it. There is then again a canonical inclusion

$$
  \partial_v : T/v \to T
  \,.
$$

This is called an **outer face map**.

=--

+-- {: .num_example }
###### Example

With $T$ as from example \ref{ExampleForInnerFace}, we have

$$
  T/v
   =
   \left\{
    \array{
     \searrow & & \searrow^e
     \\
      & \bullet &\to& \bullet &\to&
     \\
     \nearrow
    }
  \right\}
  \,.
$$

=--

+-- {: .num_defn }
###### Definition

Let $T$ be a corolla tree, example \ref{Corolla}.
There are $n+1$ injections of the tree $L_0$ with no vertex $|$ into the corolla with $n$ inputs.

$$
  | \hookrightarrow
  \array{
    \searrow &\cdots & \swarrow
    \\
    & \bullet
    \\
    & \downarrow
  }
  \,.
$$

These are called **corolla face maps** and counted as _outer face maps_.

=--

In terms of these face maps there is now a notion of _boundary_ and _horn_ of a tree in direct analogy to the notion of _[[boundary of a simplex]]_ and _[[horn]]_ in a simplex.

+-- {: .num_defn }
###### Definition

Let $T$ be a tree and let $Faces(T) \subset \Omega_{/T}$ be the set of all its face maps $\partial_\alpha \Omega[T] \to \Omega[T]$ , as defined above. The **boundary** of $T$ is the dendroidal set

$$
  \partial \Omega[T] := \coprod_{\alpha \in Faces(T)} \partial_\alpha \Omega[T]
$$

given by the [[union]] of all these faces in $\Omega[T]$.

For given $\alpha \in Faces(T)$, the $\alpha$-**horn** $\Lambda^\alpha[T] \in dSet$ of $T$ is the union of
all faces except this one:

$$
  \Lambda^\alpha[T]
   :=
  \coprod_{(\beta \neq \alpha) \in Faces(T)} \partial_\beta \Omega[T]
  \,.
$$

A horn is called an **outer horn** or an **inner horn** depending on whether the omitted face is outer or inner, respectively.

The inner horn corresponding to the inner face map given by contraction an edge $e$ is canonically denoted $\Lambda^a \Omega[T]$.

=--

These definitions are due to ([Weiss (thesis)](#WeissThesis)) and
([MoerdijkWeiss](#MoerdijkWeiss)).

+-- {: .num_remark }
###### Remark

This is a genuine generalization of the notion of horns and boundaries of simplices. The outer/inner horns of the $n$-[[simplex]] $\Delta[n]$ are taken by $i_! : sSet \to dSet$ precisely to the outer/inner hors of the linear tree $L_n = i_!(\Delta^n)$.

=--

+-- {: .num_defn }
###### Definition

For $\Lambda^\alpha \Omega[T] \to \Omega[T]$ a horn inclusion and for $X \in dSet$ a dendroidal set, an **$\alpha$-horn in $X$** is a morphism of simplicial set

$$
  \Lambda^\alpha \Omega[T] \to X
  \,.
$$

A **filler** of this horn in $X$ is an extension $\sigma$ in

$$
  \array{
    \Lambda^\alpha \Omega[T] &\to& X
    \\
    \downarrow & \nearrow
    \\
    \Omega[T]
  }
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

Choices of dendroidal _inner horn_ fillers correspond to _choices of composites_ of operations.

A dendrex $\Omega[T] \to X$ encodes a collection of operations and choices of theor composite in $X$.

An inner horn $\Lambda^e \Omega[T] \to X$ encodes a choice of operations in $X$ and their composites _except_ a choice for the composition of operations along $e$. Picking a filler for this inner horn is picking such a choice.

The outer horn fillers have different interpretation. They correspond to choices of a) inverses of linear operations 2) invertible elements of $n$-ary operation.

For more on this see _[[model structure on dendroidal sets]]_.

=--

### Skeletal filtration
  {#SkeletalFiltration}

+-- {: .num_defn #SkeletalFiltration}
###### Definition

For $X \in dSet$ a dendroidal set and $n \in \mathbb{N}$, write
$Sk_n(X) \in dSet$ for the dendroidal set which is generated from the
non-degenerate $T$-dendrices for ${\vert T\vert} \leq n$.

The sequence of inclusions

$$
  Sk_0(X) \hookrightarrow Sk_1(X) \hookrightarrow Sk_2(X) \hookrightarrow
  \cdots \hookrightarrow X
$$

is called the **skeletal filtration** of $X$.

=--

### Normal monomorphisms and normal dendroidal sets
 {#NormalMonomorphisms}

A key fact in the theory of [[simplicial sets]] is that the [[monomorphisms]] there are generated under [[pushout]], [[transfinite composition]] and [[retracts]] from the [[boundary of a simplex|boundary inclusions]] (indeed the [[model structure on simplicial sets]] is a [[cofibrantly generated model category]] with generating cofibrations the boundary inclusions).

For dendroidal sets the boundary inclusions turn out not to generate all monomorphisms, but just a subclass called the _normal_ monomorphisms. We discuss now the definition and some basic propoerties of normal monomorphisms. Most of these are a specialization of the general notion of [normal morphisms over a generalized Reedy category](http://ncatlab.org/nlab/show/generalized+Reedy+category#NormalMorphisms) to the [[generalized Reedy category]] $\Omega$.

+-- {: .num_lemma }
###### Lemma (face maps are the monomorphisms)

The inner and outer face morphisms $\partial_e$ and $\partial_v$ are precisely the [[monomorphism]]s in the [[tree category]] $\Omega$.

=--

+-- {: .proof}
###### Proof

Lemma 3.1 in [MoeWei07](http://cat.inist.fr/?aModele=afficheN&cpsidt=20087314).

=--

The **boundary of a tree** is the union of all its face dendroidal sets

$$
  \partial \Omega[T] = \cup_{e \in Edges(T)} \Omega[\partial_e T] \cup_{v \in Vertices(T)} \Omega[\partial_v T]
  \,.
$$

Compare to [[boundary of a simplex]].

Analogously then to the notion of [[horn]] of a [[simplex]], for $d$ an edge of $T$ the union

$$
  \Lambda^e \Omega[T] = \cup_{e \neq d \in Edges(T)} \Omega[\partial_e T] \cup_{v \in Vertices(T)} \Omega[\partial_v T]
$$

of dendroidal sets is the **inner horn** of $T$ at $e$, and for $w$ an outer face the union

$$
  \Lambda^w \Omega[T] = \cup_{e \neq d \in Edges(T)} \Omega[\partial_e T] \cup_{v \neq w \in Vertices(T)} \Omega[\partial_v T]
$$

is the **outer horn** at $w$.

We have the canonical **boundary inclusions**

$$
  \partial \Omega[T] \to \Omega[T]
$$

and **horn inclusions**

$$
  \Lambda^\alpha \Omega[T] \to \Omega[T]
  \,.
$$

+-- {: .num_defn #NormalMorphism}
###### Definition

A [[monomorphism]] $X \to Y$ of dendroidal sets is called **normal** if for any tree $T$, any non-degenerate dendrex $y \in Y(T)$ which does not belong to the image of $X(T)$ has a trivial [[stabilizer subgroup]] $Aut(T)_y \subset Aut(T)$ of the [[automorphism group]] of $T$.

A dendroidal set $X$ is called **normal** if $\emptyset \hookrightarrow X$ is a normal monomorphism.

=--

+-- {: .num_remark}
###### Remark

This is unrelated to the notion of _[[normal monomorphism]]_ in a context with [[zero morphisms]].

=--

Here are some equivalent characterizations of normality.

+-- {: .num_prop}
###### Proposition

A dendroidal set $X$ is normal precisely if
for every $n \in \mathbb{N}$ the canonical [[commuting diagram]]

$$
  \array{
    \coprod_{[\Omega[T] \stackrel{nd}{\to} X]} \partial \Omega[T]
    &\to&
    Sk_n(X)
    \\
    \downarrow && \downarrow
    \\
    \coprod_{[\Omega[T] \stackrel{nd}{\to} X]} \Omega[T]
    &\to&
    Sk_{n+1}(X)
  }
$$

is a [[pushout]] diagram, where $\{Sk_n(X)\}$ is the skeletal filtration, def. \ref{SkeletalFiltration} and where the coproduct is over [[isomorphism classes]] of non-degenerate dendrices.

=--

+-- {: .num_prop}
###### Proposition

A monomorphism $f : X \to Y$ in [[dSet]] is normal precisely if for every $T \in \Omega$ the [[action]] of the [[automorphism group]] $Aut(T)$ on $Y(T)-X(T)$ is a [[free action]].

Accordingly, a dendroidal set $Y$ is normal precisely if for every $T \in \Omega$ the action of $Aut(T)$ on $Y(T)$ is [[free action|free]].

=--

This is  ([CisMoer09, prop 1.5](#CisMoer09)).

+-- {: .num_cor}
###### Corollary

For $f : X \to Y$ any morphism between dendroidal sets, then

* if $Y$ is normal, then $X$ is normal;

* if $Y$ is normal and $f$ is [[monomorphism|monic]], then $f$ is normal.

=--

+-- {: .num_example}
###### Example

* For every tree $T$, the dendroidal set $\Omega[T]$ is normal.

* For every [[simplicial set]], the dendroidal set $i_!(X)$ is normal.

=--

+-- {: .num_prop}
###### Proposition

The class of normal morphisms in [[dSet]] is generated from the boundary inclusions under

* [[pushouts]];

* [[transfinite compositions]];

* [[retracts]].

In particular it is closed under these operations.

=--

This is  ([CisMoer09, prop 1.4](#CisMoer09)).

### Boardman-Vogt tensor product
 {#BoardmanVogtTensorProduct}

As any [[category of presheaves]], [[dSet]] is a [[cartesian monoidal category]]. However, the cartesian [[tensor product]] is not the natural one with respect to the inclusion of [[operads]] into dendroidal sets. The natural monoidal structure on [[Operad]] is rather a generalization of the [[Boardman-Vogt tensor product]].

+-- {: .num_defn #BVTensorProduct}
###### Definition

Write

$$
  N_d(-\otimes -)
  :
  \Omega \times \Omega
  \hookrightarrow
  Operad \times Operad
  \hookrightarrow
  Operad
  \stackrel{N_d}{\to}
  dSet
$$

for the functor that forms the [[Boardman-Vogt tensor product]] of the free operads given by two trees, and then regards the result as a dendroidal set by the dendroidal nerve, def. \ref{DendroidalNerve}.

The **tensor product of dendroidal sets** is the [[Yoneda extension]]

$$
  -\otimes - : dSet \times dSet \to dSet
$$

of this functor, hence the unique such functor which preserves [[colimits]] in both variables and coincides with the BV-tensor product of operads on $\Omega$.

=--

With respect to this tensor product is [[dSet]] a colax symmetric monoidal category ( [Theorem 6.3.4 of HHM13](#HeutsHinichMoerdijk13)). This is discussed [below](#ClosedMonoidalStructure).

+-- {: .num_prop}
###### Proposition

For $A \to B$ and $X \to Y$ in [[dSet]] two normal monomorphisms, def. \ref{NormalMorphism},
the canonical morphism out of their [[pushout-product axiom|pushout product]]

$$
  (A \otimes Y) \coprod_{A \otimes X} (B \otimes X)
  \to
  B \otimes Y
$$

is also normal.

=--

This is ([CisMoer09, prop 1.9](#CisMoer09)).

## Properties

### Relation to simplicial sets
  {#RelationToSimplicialSets}

Write $0$ for the tree consisting of a single edge. Let $\Omega/0$ be the [[over category]]. By inspection one sees that

+-- {: .num_prop }
###### Proposition

There is an [[equivalence of categories]]

$$
  \Omega/0 \simeq \Delta
$$

of the slice with the [[simplex category]].

Moreover, the over-category of all dendroidal sets over $\eta := \Omega[0]$ is the category [[sSet]] of [[simplicial set]]s

$$
  dSet/\Omega[0] \simeq sSet
  \,.
$$

=--

By general properties of [[Kan extension]] we have the following

+-- {: .num_prop }
###### Proposition

The corresponding canonical [[forgetful functor]]

$$
  i : \Delta \stackrel{\simeq}{\to} \Omega/0 \to \Omega
$$

is

1. a [[full and faithful functor]];

1. a [[sieve]].

=--

+-- {: .num_prop }
###### Proposition

The functor $i : \Delta \to \Omega$ induces an [[adjoint triple]]

$$
  (i_! \dashv i^* \dashv i_*)
   :
   sSet
  \stackrel{
    \overset{i_!}{\hookrightarrow}
  }{
    \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}
  }
  dSet
  \,,
$$

where

* $i^*$ is given by precomposition with $i$;

* $i_!$ is a [[full and faithful functor]] defined equivalently as

  * the [[left adjoint]] to $i^*$;

  * the left [[Kan extension]] of $i$;

  * the canonical functor $sSet \stackrel{\simeq}{\to} dSet/\Omega[0] \to dSet$;

  * the functor that sends  $X \in sSet$ to

    $$
      i_!(X)
       :
     T
     \mapsto
      \left\{
        \array{
          X_n & if\,T\,is\,linear\,with\,n\,vertices;
          \\
          \emptyset & otherwise
        }
      \right.
      \,.
    $$

Hence $i_!$ is a [[full and faithful functor]], and hence (see the discussion at [[adjoint triple]]) so is $i_*$.

Accordingly, $i : sSet \hookrightarrow dSet$ is an [[open geometric morphism|open]] [[geometric embedding]] of [[presheaf toposes]].

=--

+-- {: .num_remark }
###### Remark

The dendroidal set $i_!([n])$ that corresponds to the $n$-[[simplex]] is the linear tree; example \ref{LinearTree}, with $(n+1)$-edges:

$$
  i([n])
  =
  \left(
  \stackrel{0}{\to}
    \bullet
  \stackrel{1}{\to}
    \bullet
  \stackrel{2}{\to}
  \cdots
  \bullet
  \stackrel{n}{\to}
  \right)
  \,.
$$

We think of each bullet as a unary operation -- an ordinary [[morphism]] -- $\stackrel{n}{\to}\bullet \stackrel{n+1}{\to}$ from the [[object]] $n$ to the object $(n+1)$.

Notice that this is hence Poincaré-dual to how one tends to visualize $[n]$ itself

$$
  [n] = (0 \to 1 \to 2 \to \cdots \to n)
$$

and how one tends to visualize morphisms $n \to (n+1)$.

=--

The embedding of simplicial sets into dendroidal sets compatibly relates the [[cartesian monoidal category|Cartesian monoidal structure]] $(sSet, \times)$ with the [Boardman-Vogt monoidal structure](#BoardmanVogtTensorProduct) $(dSet, \otimes_{BV})$

+-- {: .num_prop }
###### Proposition

The functor

$$
  i_! : (sSet, \times) \to (dSet, \otimes_{BV})
$$

is a [[strong monoidal functor]], in that for all $X, Y \in sSet$
there is a [[natural isomorphism]]

$$
  i_!(X \times Y) \simeq i_!(Y) \otimes_{BV} i_!(Y)
$$

in [[dSet]].

Moreover, $(i_! \dashv i^*)$ respects the [[internal hom]] of dendroidal sets, prop. \ref{TheClosedMonoidalStructure}, in that for all $X,Y \in sSet$ and $D \in dSet$

* $i^* [i_!(X), D]_{BV} \simeq [X, i^*(D)]$;

and hence in particular

* $i^* [i_!(X), i_!(Y)]_{BV} \simeq [X, Y]$.

=--

This appears as ([Moerdijk-Weiss, prop. 5.3](#MoerdijkWeiss)).

### Relation to $Set$-operads -- dendroidal nerve
 {#RelationToOperads}

By the general notion of [[nerve and realization]], the
inclusion $\Omega \hookrightarrow Operad$ from def. \ref{FreeOperadCategory}
induces a [[nerve]] operation on [[operads]] with values in
[[simplicial sets]].

+-- {: .num_defn #DendroidalNerve}
###### Definition

For $O \in Operad$ an [[operad]], its **[[dendroidal nerve]]**
is the dendroidal set given by

$$
  N_d(O) : T \mapsto Hom_{Operad}(T, O)
  \,.
$$

This extends to a [[functor]]

$$
  N_d : Operad \to dSet
  \,.
$$

=--

By general properties of [[nerve and realization]] we have:

+-- {: .num_prop}
###### Proposition

The dendroidal nerve has the following properties.

* It extends the simplicial nerve of operads in that

  $$
    i^* \circ N_d = N
    \,.
  $$

* It has a [[left adjoint]]

  $$
    \tau_d : dSet \to Operad
  $$

  given by [[left Kan extension]].

* $N_d$ is a [[full and faithful functor]], equivalently there is a [[natural isomorphism]]

  $$
    \tau_d \circ N_d \simeq id
    \,.
  $$

=--

See for instance ([Moerdijk-Weiss, section 4](#MoerdijkWeiss)).

For $X$ a dendroidal set, $\tau_d(X)$ is also called the **operad generated by** $X$.

In conclusion we find [[reflective subcategory]]

$$
  (\tau_d \dashv N_d)
   :
  Operad
    \stackrel{\overset{\tau_d}{\leftarrow}}{\underset{N_d}{\hookrightarrow}}
  dSet
  \,.
$$

This is compatible with the [[Boardman-Vogt tensor product]] as defined [above](#BoardmanVogtTensorProduct):

+-- {: .num_prop}
###### Proposition

The functor $\tau_d : (dSet, \otimes) \to (Operad, \otimes_{BV})$ is a [[strong monoidal functor]], in that for all $X, Y \in dSet$ there is a [[natural isomorphism]]

$$
  \tau_d(X \otimes Y ) \simeq \tau_d(X) \otimes_{BV} \tau_d(Y)
  \,.
$$

=--

This appears as ([Moerdijk-Weiss, prop. 5.2](#MoerdijkWeiss)).

+-- {: .num_remark}
###### Remark

Since $\tau_d N_d \simeq id$ in particular we have for all $P, Q \in Operad$
an isomorphism

$$
  \tau_d (N_d(P) \otimes N_d(Q)) \simeq P \otimes_{BV} Q
  \,.
$$

=--

### Relation to $sSet$-operads -- homotopy coherent dendroidal nerve
 {#HomotopyCoherentNerve}

The dendroidal nerve of [[Set]]-[[operads]] discussed
[above](#RelationToOperads) is important for setting up the model of
dendroidal sets. But the usefulness of the model comes from
its relation to [[sSet]]/[[Top]]-enriched operads ([[topological operads]])
via an operadic generalization of the [[homotopy coherent nerve]] that sends [[sSet-enriched categories]] to [[simplicial sets]].

Write $Operad_sSet$ for the category of [[sSet]]-enriched operads ([[simplicial operads]]). Write

$$
  W_H : Operad_sSet \to Operad_sSet
$$

for the [[Boardman-Vogt resolution]] functor.

The restriction of this along the inclusion of free Set-operads

$$
  \Omega \hookrightarrow Operad
  \hookrightarrow
  Operad_{sSet}
  \stackrel{W_H}{\to}
  Operad_{sSet}
  \,,
$$

which will also be denoted just $W_H$ in the following,
canonically induces [[nerve and realization]] functors:

+-- {: .num_defn }
###### Definition

The **[[dendroidal homotopy coherent nerve]]** of a [[simplicial operad]] $P$ is the dendroidal set $hcN_d(P)$ given by

$$
  hcN_d(P) : T \mapsto Hom_{Operad_{sSet}}(W_H(T), P)
 \,.
$$

This extends to a [[functor]]

$$
  hcN_d : Operad_{sSet} \to dSet
  \,.
$$

=--

+-- {: .num_defn }
###### Definition

We write

$$
  {\vert -\vert}_H : dSet \to Operad_{sSet}
$$

(or $W_!$ as [[model structure on dendroidal sets|here]])
for the corresponding [[left adjoint]] [[nerve and realization|realization]].

=--

Via this [[adjunction]]

$$
  ({\vert -\vert}_H \dashv hcN_d)
  :
  Operad_{sSet}
  \to dSet
$$

we may understand generally the theory of dendroidal sets as being about BV-resolutions of simplicial operads.

+-- {: .num_prop }
###### Proposition

Let $P \in Operad_{sSet}$.

There is an [[isomorphism]] of simplicial operads

$$
  W_H(P) \stackrel{\simeq}{\to}
  {\vert hcN_d P \vert}_H
$$

between the [[Boardman-Vogt resolution]] $W_H(P)$ of $P$ and the homotopy-realization of its homotopy-coherent dendroidal nerve.

=--

All of this is an operadic generalization of the _[[relation between quasi-categories and simplicial categories]]_.

### The full diagram of relations
 {#FullDiagram}

The above functors between dendroidal sets and
simplicial sets ([here](#RelationToSimplicialSets))
and operads ([here](#RelationToOperads))
arrange into the following [[diagram]]

$$
  \array{
     sSet
     &\stackrel{\overset{i_!}{\to}}{\underset{i^*}{\leftarrow}}&
     dSet
     \\
     {}^\mathllap{\tau} \downarrow
     \uparrow^{\mathrlap{N}}
     &&
     {}^{\mathllap{\tau_d}}
     \downarrow
     \uparrow^{N_d}
     \\
     Cat
     &
     \stackrel{\overset{j_!}{\to}}{\underset{j^*}{\leftarrow}}
     &
     Operad
  }
  \,,
$$

where functors on the left and on top are [[left adjoint|left adjoints]] to those on the right and on the bottom, respectively.

This commutes in three of four possible ways, up to [[natural isomorphism]]
in that

1. $j_! \tau \simeq \tau_d i_!$;

1. $N j^* \simeq i^* N_d$;

1. $i_! N \simeq N_d j_!$.

There is also a [[natural transformation]]

$$
  \tau i^* \Rightarrow j^* \tau_d
  \,,
$$

but not all of its components are isomorphisms.

Moreover, $N, N_d$,$i_!, j_!$ are [[full and faithful functors]]
and hence (see the properties of [[adjoint functors]])

1. $\tau N \simeq id$;

1. $\tau_d N_d \simeq id$;

1. $i^* i_! \simeq id$;

1. $j^* j_! \simeq id$.

## Structure on $dSet$ ##

Some [[mathematical structure|structure]] carried by the [[category]] [[dSet]] of [[dendroidal sets]]:

### Homotopical monoidal structure
 {#ClosedMonoidalStructure}

[Heuts-Hinich-Moerdijk 13, section 6.3](#HeutsHinichMoerdijk13)

### $sSet$-enriched structure

Using the fact that [[dSet]] is a [[closed monoidal category]] with [[internal hom]] dendroidal sets $[C,D]$ for dendroidal sets $C$ and $D$, and using the functor $i^* : dSet \to SSet$ we obtain canonically the structure of an [[simplicially enriched category]] / [[sSet]]-[[enriched category]] on $dSet$ with the hom-simplicial set between $C$ and $D$ being $i^*[C,D]$.

### Model category structure

The category $dSet$ carries the Cisinski-Moerdijk [[model structure on dendroidal sets]]. With this model structure it forms a [[monoidal model category]].

Together with the fact that $i^*: dSet \to sSet$ is a [[right Quillen functor]] (with respect to the [[model structure for quasi-categories]]) this imples that [[dSet]] is an $sSet_{Joyal}$-[[enriched model category]] (but not, without further work, an $sSet_{Quillen}$-enriched model category!).

## Related concepts

* [[dSet]]

* [[model structure on dendroidal sets]]

* [[Cartesian fibration of dendroidal sets]]

* [[minimal dendroidal fibration]]

* [[simplicial set]], [[globular set]], [[cubical set]], [[cellular set]]

## References
 {#References}

Textbook account:

* [[Gijs Heuts]], [[Ieke Moerdijk]], _Simplicial and Dendroidal Homotopy Theory_, Ergebnisse der Mathematik und ihrer Grenzgebiete **75**, Springer (2022) [ISBN 9783031104473, 9783031104473, [doi:10.1007/978-3-031-10447-3](https://doi.org/10.1007/978-3-031-10447-3)]

Surveys:

* [[Ieke Moerdijk]], _Lectures on dendroidal sets_ , lectures given at the Barcelona workshop on _[Simplicial methods in higher categories](http://www.crm.es/HigherCategories/)_ (2008) ([preliminary writeup](http://www.crm.cat/HigherCategories/hc1.pdf))

* {#Weiss} [[Ittay Weiss]], _From operads to dendroidal sets_ , in _[[Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, AMS (2011)

Dendroidal sets were introduced in

* {#MoerdijkWeiss}  [[Ieke Moerdijk]] [[Ittay Weiss]], _Dendroidal sets_,  Algebraic & Geometric Topology 7 (2007) 1441–1470,  ([doi:10.2140/agt.2007.7.1441](http://msp.warwick.ac.uk/agt/2007/07/p056.xhtml), [arXiv:math.AT/0701293](http://arxiv.org/abs/math.AT/0701293))

A discussion of dendroidal inner Kan complexes (see also at _[[model structure on dendroidal sets]]_) appeared in

* [[Ieke Moerdijk]], [[Ittay Weiss]], _On inner Kan complexes in the category of dendroidal sets_, [math.CT/0701295](http://arxiv.org/abs/math/0701295)

The thesis

* [[Ittay Weiss]], _Dendroidal sets_ PhD thesis ([web](http://igitur-archive.library.uu.nl/dissertations/2007-0918-200833/UUindex.html))
 {#WeissThesis}

contains essentially the material of these two articles, together with a discussion of broad posets.

The [[model structure for dendroidal complete Segal spaces]] and the [[model structure for Segal operads]] was constructed in

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal Segal spaces and infinity-operads_, ([arxiv:1010.4956](http://arxiv.org/abs/1010.4956))

Normal morphisms of dendroidal sets are discussed for instance around prop. 1.4 of

* {#CisMoer09} [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets as models for homotopy operads_ ([arXiv:0902.1954](http://arxiv.org/abs/0902.1954))

See also

* {#HeutsHinichMoerdijk13} Gijs Heuts, Vladimir Hinich, Ieke Moerdijk, _On the equivalence between Lurie's model and the dendroidal model for infinity-operads_ ([arXiv:1305.3658](https://arxiv.org/abs/1305.3658))

[[!redirects dendroidal sets]]

[[!redirects dendroidal nerve]]


denial inequality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

\tableofcontents

## Idea

A **weak inequality relation** or **denial inequality relation** $\neq$ on a set $S$ is the [[negation]] of the [[equality]] relation on $S$.

$$a \neq b \coloneqq \neg(a = b)$$

This is a term found in [[constructive mathematics]], to distinguish from [[strict inequality relation]] or other notions of [[inequality relation]] such as [[tight apartness relations]]. In [[classical mathematics]], weak inequality relations and [[strict inequality relations]] are the same and are simply called "inequality", so the notion plays no further role classically.

If one wishes to reserve the word "inequality" for [[order]] relations (such as $\le$ and $\lt$), then one may instead use the word **disequality** to refer to the weak inequality.  (For instance, this is common in [[type theory]] with [[subtype]] relations that form an ordering on the [[types]].)

## Properties

The weak inequality relation is an [[irreflexive symmetric relation]]. Irreflexivity follows from the negation of reflexivity of [[equality]], while symmetry follows from the [[contrapositive]] of symmetry of equality. In addition, the weak inequality relation is a [[stable relation]], because in [[constructive mathematics]], given any proposition $P$, $\neg \neg \neg P$ implies that $\neg P$. This means that $\neg \neg \neg (a = b)$ implies that $\neg (a = b)$, and since $\neg (a = b)$ if and only if $a \neq b$, this means that $\neg \neg (a \neq b)$ implies that $a \neq b$. Weak inequality is a [[weakly tight relation]], in that for all elements $a \in A$ and $b \in A$, $\neg (a \neq b)$ implies that $\neg \neg (a = b)$, because by definition, $\neg (a = b)$ if and only if $a \neq b$, and so by contraposition, $\neg \neg (a = b)$ if and only if $\neg (a \neq b)$.

### Denial apartness

The denial inequality relation on a set $S$ is said to be a **denial apartness relation** if for all elements $a \in S$, $b \in S$ and $c \in S$, $\neg((a \neq b) \wedge (b \neq c))$ implies that $\neg(a \neq b) \vee \neg(b \neq c)$.

This implies that $\neq$ is an [[apartness relation]] because the contrapositive of the transitive property of equality states that $a \neq c$ implies that $\neg((a = b) \wedge (b = c))$, and the above condition implies that $\neg(a \neq b) \vee \neg(b \neq c)$, which is precisely the condition for a [[comparison]].

In particular, if [[de Morgan's law]] holds in the logic, then every weak inequality relation is a denial apartness relation.

Similarly, every [[stable relation|stable]] [[tight apartness relation]] is a denial apartness relation.

## See also

* [[stable relation]]

* [[inequality relation]]

* [[tight inequality relation]]

* [[stable equality]]

* [[Kock field]]

[[!redirects not equal]]
[[!redirects not equal to]]

[[!redirects weak inequality]]
[[!redirects weak inequalities]]

[[!redirects weak inequality relation]]
[[!redirects weak inequality relations]]

[[!redirects denial inequality]]
[[!redirects denial inequalities]]

[[!redirects denial inequality relation]]
[[!redirects denial inequality relations]]

[[!redirects denial apartness]]

[[!redirects denial apartness relation]]
[[!redirects denial apartness relations]]

[[!redirects disequality]]


Denis Auroux

[[!redirects D. Auroux]]
Denis Auroux is a professor in the Department of Mathematics,
University of California, Berkeley.

* [webpage](http://math.berkeley.edu/~auroux/)


Denis Diderot

* [Wikipedia entry](https://en.wikipedia.org/wiki/Denis_Diderot)

On the need for his _[Encyclopédie](http://quod.lib.umich.edu/cgi/t/text/text-idx?c=did;view=text;rgn=main;idno=did2222.0000.004)_ from 1750 (see [here](http://www.historyofinformation.com/expanded.php?id=2877)):

> As long as the centuries continue to unfold, the number of books will grow continually, and one can predict that a time will come when it will be almost as difficult to learn anything from books as from the direct study of the whole universe. It will be almost as convenient to search for some bit of truth concealed in nature as it will be to find it hidden away in an immense multitude of bound volumes. When that time comes, a project, until then neglected because the need for it was not felt, will have to be undertaken.

> $[...]$

> Thanks to encyclopedic ordering, the universality of knowledge, and the frequency of references, the connections grow, the links go out in all directions, the demonstrative power is increased, the word list is complemented, fields of knowledge are drawn closer together and strengthened; we perceive either the continuity or the gaps in our system, its weak sides, its strong points, and at a glance on which objects it is important to work for one's own glory, or for the greater utility to humankind. If our dictionary is good, how many still better works it will produce.

## related $n$Lab entries

* [[schreiber:What is... the nLab]]

[[!redirects Diderot]]

category: people


Denis Nardin

* [personal page](https://homepages.uni-regensburg.de/~nad22969/)

## Selected writings

On [[stable homotopy theory]]:

* [[Denis Nardin]], _Introduction to stable homotopy theory_, Regensburg 2021 ([webpage](https://homepages.uni-regensburg.de/~nad22969/stable-homotopy-theory2020.php), [pdf](https://homepages.uni-regensburg.de/~nad22969/stable-homotopy-2020/stable-homotopy.pdf), [video](https://www.youtube.com/watch?v=v5Rr_Z8QsOs&list=PLQuS12Ap6va-VgI5cdIjdHquIH3sLCLdJ))

## Related $n$Lab entries

* [[Parametrized Higher Category Theory and Higher Algebra]]

category: people


Denis O'Brien

* [IEEE page](https://ieeexplore.ieee.org/author/38241569400)

## Selected writings

Observation that [[vacua]]/[[ground states]] of the [[free field|free]] [[Dirac field]] in a [[electromagnetic field|Coulomb]] [[background field]] are characterized by [[Fredholm operators]] whose [[index]] is the total charge in the vacuum:

* {#CareyHurstOBrien82} [[Alan Carey]], [[Charles Angas Hurst]], [[Denis O'Brien]], *Automorphisms of the canonical anticommutation relations and index theory*, Journal of Functional Analysis **48** 3 (1982) 360-393 $[$doi:10.1016/0022-1236(82)90092-1$]$

category: people


Denis Sjerve

* [webpage](http://www.math.ubc.ca/~sjer/)

## Selected writings

On the [[rational cohomology]] of [[iterated loop spaces]] of [[n-spheres]] ([[cocycle spaces]] in [[rational Cohomotopy]] on [[n-spheres]]):

* {#KallelSjerve99} [[Sadok Kallel]], [[Denis Sjerve]], _On Brace Products and the Structure of Fibrations with Section_, 1999 ([pdf](https://www.math.ubc.ca/~sjer/brace.pdf), [[KallelSjerv99.pdf:file]])

## Related $n$Lab entries

* [[iterated loop space]]

* [[rational Cohomotopy]]

category: people


Denis-Charles Cisinski

* [website](https://cisinski.app.uni-regensburg.de/)

* [[Denis-Charles Cisinski]], _Catégories supérieures et théorie des topos_, Séminaire Bourbaki, 2015-03-21, [pdf](https://cisinski.app.uni-regensburg.de/1097.pdf).

## Selected writings

On [[(∞,1)-category theory]] and [[homotopical algebra]]:

* [[Denis-Charles Cisinski]], *Faisceaux localement asphériques* (2003) [[pdf](https://cisinski.app.uni-regensburg.de/mtest2.pdf), [[Cisinski-FaisceauxLocAsph.pdf:file]]]

* [[Denis-Charles Cisinski]], _Les préfaisceaux comme modèles des types d'homotopie_, Astérisque 308, 2006 ([doi:10.24033/ast.715](https://smf.emath.fr/publications/les-prefaisceaux-comme-modeles-des-types-dhomotopie), [NUMDAM](http://www.numdam.org/item/AST_2006__308__R1_0/))

* [[Denis-Charles Cisinski]], _[[Higher categories and homotopical algebra]]_, Cambridge University Press, 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](https://cisinski.app.uni-regensburg.de/CatLR.pdf))

On [[triangulated categories]] of [[mixed motives]] and their [[yoga of six functors]]:

* {#CisinskiDeglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]

and the relevant [[model structures on chain complexes]]:

* [[Denis-Charles Cisinski]], [[Frédéric Déglise]], *Local and stable homological algebra in Grothendieck abelian categories*, Homology, Homotopy and Applications **11** 1 (2009) 219–260 [[arXiv:0712.3296](http://arxiv.org/abs/0712.3296), [hha:1251832567](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-11/issue-1/Local-and-stable-homological-algebra-in-Grothendieck-abelian-categories/hha/1251832567.full)]

On the [[(infinity,1)-Grothendieck construction]] via [[straightening and unstraightening]] entirely within the context of [[quasi-categories]]:

* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]

## Related $n$Lab entries

* [[test category]], [[test topos]]

* [[derivator]]

* [[Cisinski model structure]]

* [[motives]]
  * [[mixed motives]]
  * [[category of motives]]
  * [[motivic homotopy theory]]
  * [[motivic cohomology]]
  * [[algebraic K-theory]]
  * [[noncommutative motives]]

* [[(∞,1)-operad]]s
  * [[model structure on dendroidal sets]]

* [[homotopy groups in an (∞,1)-topos]]

* [[univalence axiom]]

category: people

[[!redirects Cisinski]]


Deniz Sarikaya

* [website](https://www.math.uni-hamburg.de/spag/ml/Sarikaya/)

## Selected publications

* [[Stefania Centrone]], [[Deborah Kant]], [[Deniz Sarikaya]] (editors), *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

category: people


Dennis Borisov

* [webpage](http://sites.google.com/site/dennisborisov/)

category: people


Dennis Dieks

* [Wikipedia entry](http://en.wikipedia.org/wiki/Dennis_Dieks)

## related $n$Lab entries

* [[no-cloning theorem]]

category: people


Dennis Gaitsgory

__Dennis Gaitsgory__ is a mathematician at Harvard. His research is in representation theory and algebraic geometry, specially D-modules, chiral algebras and geometric Langlands.

* [webpage](http://www.math.harvard.edu/~gaitsgde)

## Selected writings

On the [[geometric Langlands conjecture]] ([[categorical geometric Langlands conjecture]]):

* [[Dennis Gaitsgory]], _[[Notes on geometric Langlands]]_, [web](http://www.math.harvard.edu/~gaitsgde/GL/).

* [[D. Gaitsgory]], [[N. Rozenblyum]], _[[A study in derived algebraic geometry]]_.

* Informal notes on the [[Fargues-Fontaine curve]] with [[Jacob Lurie]].

On the [[quantum geometric Langlands correspondence]]:

* [[Dennis Gaitsgory]], *Quantum Langlands Correspondence* [[arXiv:1601.05279](https://arxiv.org/abs/1601.05279)]

[[!redirects D. Gaitsgory]]


Dennis Perepelitsa

* [webpage](http://phys.columbia.edu/~dvp/)

* [previous webpage](http://web.mit.edu/dvp/www/)

## related $n$Lab entries

* [[baryogenesis]]

category: people


Dennis Sullivan

Dennis Sullivan is an American topologist. His initial work was in geometric topology, but later he developed theories of localisation of homotopy types, rational homotopy theory, and aspects of string topology.

The editor ([[Andrew Ranicki]]) of the redistribution of his 1970 notes wrote:

The notes had a major influence on the development of both algebraic and geometric topology, pioneering

*  the localization and completion of spaces in homotopy theory, including p-local, profinite and rational homotopy theory, leading to the solution of the [[Adams conjecture]] on the relationship between vector bundles and spherical fibrations,

* the formulation of the 'Sullivan conjecture' on the contractibility of the space of maps from the classifying space of a finite group
to a finite dimensional CW complex,

* the action of the Galois group of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$ on smooth manifold
structures in profinite homotopy theory,

* the K-theory orientation of [[PL manifold]]s and bundles.

## External links

* [website](http://www.math.sunysb.edu/~dennis/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Dennis_Sullivan)

* interview by [[Kathryn Hess]]: [video](https://www.youtube.com/watch?v=K8kPS6FZBoc&feature=youtu.be)

## Selected writings

On [[localization of a space|localization in homotopy theory]]:

* Dennis Sullivan, [Geometric Topology: Localization, Periodicity, and Galois Symmetry (The 1970 MIT notes)](http://www.maths.ed.ac.uk/~aar/books/gtop.pdf)

Introducing [[string topology]]:

*  {#ChasSullivan} [[Moira Chas]], [[Dennis Sullivan]], _String topology_ [math.GT/9911159](http://arxiv.org/abs/math/9911159)

*  [[Ralph Cohen]], John R. Klein, [[Dennis Sullivan]], _The homotopy invariance of the string topology loop product and string bracket_, J. of Topology 2008 __1__(2):391-408; [doi](http://dx.doi.org/10.1112/jtopol/jtn001)

Exposition of the perspective of regarding [[string topology]]-operations as the [[TQFT]] of a [[topological string]] [[sigma model]]:

* [[Dennis Sullivan]], _Sigma models and string topology_, in: Mikhail Lyubich, Leon Takhtajan (eds.), _Graphs and Patterns in Mathematics and Theoretical Physics_, Proc. Symp. Pure Math. 73 (2005) ([doi:10.1090/pspum/073](https://doi.org/10.1090/pspum/073), [spire:1697823](http://inspirehep.net/record/1697823))

* [[Dennis Sullivan]], _Open and closed string field theory interpreted in classical algebraic topology_, chapter 11 in: [[Ulrike Tillmann]] (ed.) _Topology, Geometry and Quantum Field Theory_, Cambridge University Press (2005) ([doi:10.1017/CBO9780511526398.014](https://doi.org/10.1017/CBO9780511526398.014))

On [[homotopy theory]] and the [[Adams conjecture]]:

* Dennis Sullivan, _Genetics of homotopy theory and the Adams conjecture_, Annals of Math., 100, (1974), 1–79.

On what came to be called [[Sullivan models]]:

* [[Dennis Sullivan]], *Differential forms and the topology of manifolds*, in Proc. International  Conf.: *Manifolds Tokyo (1973)*, Univ. Tokyo Press (1975) 37-49 [[pdf](https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0017.pdf), [[Sullivan-DiffFormsManifolds.pdf:file]]]

On [[rational homotopy theory]]:

* Dennis Sullivan, _Infinitesimal computations in topology,_  Publications mathématiques de l' I.H.É.S. __47__ (1977) 269-331, [numdam](http://www.numdam.org/item?id=PMIHES_1977__47__269_0), [pdf](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1977__47_/PMIHES_1977__47__269_0/PMIHES_1977__47__269_0.pdf)

On [[Sullivan models for free loop spaces]]:

* {#VigueSullivan76} [[Micheline Vigué-Poirrier]], [[Dennis Sullivan]], _The homology theory of the closed geodesic problem_, J. Differential Geometry 11 (1976) 633-644.

On the statement that _[[the co-binary Sullivan differential is the dual Whitehead product]]_:

* {#DeligneGriffithsMorganSullivan75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent Math (1975) 29: 245 ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))

## related $n$Lab entries

* [[rational homotopy theory]]

* [[Sullivan conjecture]]

* [[model structure on dg-algebras]]

* [[string topology]]

* [[Simons-Sullivan structured bundle]]

* [[Whitehead product]], [[the co-binary Sullivan differential is the dual Whitehead product]]

category: people

[[!redirects D. Sullivan]]
[[!redirects Sullivan]]


Dennis trace

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea##

The _Dennis trace_ is a natural morphism from [[algebraic K-theory]] of rings to [[Hochschild homology]]

$$
  K(-) \longrightarrow THH(-)
  \,.
$$

While various versions of [[Chern character]] are maps from [[K-theory]] to some (co)homology of a space, algebra, scheme, the __Dennis trace__ map underlies this phenomenon at the more fundamental level of K-theory space.

The Dennis trace lifts to to [[ring spectra]] and as such factors through the map $TC(-) \longrightarrow THH(-)$ ]] from [[topological cyclic homology]], this lift is called the _[[cyclotomic trace]]_.

## Literature

* [[Ib Madsen]], _Algebraic K-theory and traces_, Current Developments in Mathematics, 1995.

* [[John Rognes]], after [[Marcel Bökstedt]], _Trace maps from the algebraic K-theory of the integers_, [K-theory archive](http://www.math.uiuc.edu/K-theory/0117)

* M. R. Kantorovitz, _Adams operations and the Dennis trace map_, JPAA __144__, 1 (Dec 1999), 21-27 <a href="http://dx.doi.org/10.1016/S0022-4049(98)00048-6"> [doi] </a>

* M. R. Kantorovitz,  C. Miller, _An explicit description of the Dennis trace map_, [claudia-revised.ps](http://www.math.uiuc.edu/~ruth/preprints/claudia-revised.ps)

* {#BlumbergGepnerTabuada11} [[Andrew Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _Uniqueness of the multiplicative cyclotomic trace_ ([arXiv:1103.3923](http://arxiv.org/abs/1103.3923))

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], _The local structure of algebraic K-theory_, Springer 2013

Relation to [[bicategorical trace]]:

* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]

[[!redirects Dennis trace map]]


Dennis Westra

* [webpage](http://www.mat.univie.ac.at/~westra/)

## related $n$Lab entries

* [[superalgebra]]

* [[supergroup]]

category: people


denotational semantics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computing
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Denotational semantics
* table of contents
{: toc}

## Idea

In [[computer science]] and [[formal logic]], *denotational semantics* refers [[semantics]] based on the idea that [[programs]] and the [[data]] they manipulate are symbolic realizations of abstract [[mathematics|mathematical]] [[objects]].

For example, the denotational semantics

*  of [[numerals]] are [[numbers]],

*  of [[programs]] are [[functions]] (approximately).

The idea of denotational semantics is thus to associate an appropriate  mathematical object, such as a number, a tuple, or a function, with each [[term]] of the given [[programming language]].

A key requirement on denotational semantics is that it respects the *compositionality* of programming languages, hence that the semantics of [[terms]] [[term introduction|constructed]] from sub-terms is correspondingly built from the semantics of these sub-terms.

{#UnderCurryHoward} Under the [[Curry-Howard correspondence]], also [[proofs]] can be seen as programs and thus one can apply denotational semantics to proofs. In this way proofs are interpreted as functions or more generally as [[morphisms]] of some [[category]]. This is used to interpret proofs of [[intuitionistic logic]] and [[linear logic]] as morphisms of respectively [[cartesian closed categories]] and [[star-autonomous categories]], as discussed at *[[relation between category theory and type theory]]*. For more general such semantics see at *[[categorical semantics of dependent type theory]]*.

## Related entries

* [[computation]]

* [[domain theory]]

* [[semantics]]

  * [[categorical semantics]]

  * [[operational semantics]]

  * [[distributional semantics]]

## References

Denotational semantics originates with the proposal of [[domain theory]] to regard [[data types]] as [[posets]] ([[(0,1)-categories]]):

* [[Dana S. Scott]], *Outline of a mathematical theory of computation*, in: Proceedings of the *Fourth Annual Princeton Conference on Information Sciences and Systems* (1970) 169–176. [[pdf](https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf), [[Scott-TheoryOfComputation.pdf:file]]]

* [[Dana S. Scott]], [[Christopher Strachey]], *Toward a Mathematical Semantics for Computer Languages*, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [[pdf](https://www.cs.ox.ac.uk/files/3228/PRG06.pdf), [[ScottStrachey-MathematicalSemantics.pdf:file]]]

* {#Scott76} [[Dana Scott]], *Data types as lattices*. SIAM Journal of Computing **5** 3 (1976) 522--587 [[doi:10.1137/0205037](https://doi.org/10.1137/0205037), [pdf](https://www.cs.ox.ac.uk/files/3287/PRG05.pdf)]

Lectures and introductions:

* [[Robert D. Tennent]], *The denotational semantics of programming languages*, Communications of the ACM **19** 8 (1976) 437–453 [[doi:10.1145/360303.360308](https://doi.org/10.1145/360303.360308), [pdf](https://dl.acm.org/doi/pdf/10.1145/360303.360308)]

* [[Robert D. Tennent]], *Denotational semantic*, in: *[[Handbook of Logic in Computer Science]]* **3**, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625?cc=de&lang=en&)]

* [[Andrew M. Pitts]], *Lecture Notes on Denotational Semantics* (2012) [[pdf](https://www.cl.cam.ac.uk/teaching/1112/DenotSem/dens-notes-bw.pdf), [[Pitts-DenotationalSemantics.pdf:file]]]

Textbook accounts:

* [[Glynn Winskel]], §5 of: *The Formal Semantics of Programming Languages*, MIT Press (1993) [[ISBN:9780262731034](https://mitpress.mit.edu/9780262731034/the-formal-semantics-of-programming-languages/), [pdf](https://www.cin.ufpe.br/~if721/intranet/TheFormalSemanticsofProgrammingLanguages.pdf)]

* [[Kenneth Slonneger]], [[Barry Kurtz]], *Denotational semantics* [[pdf](https://homepage.divms.uiowa.edu/~slonnegr/plf/Book/Chapter9.pdf)], Chapter 9 of: *Formal Syntax and Semantics of Programming Languages*, Addison-Wesley (1995) [[webpage](https://homepage.divms.uiowa.edu/~slonnegr/plf/Book/), [pdf](https://doc.lagout.org/science/Artificial%20Intelligence/Natural%20Language%20Processing/Formal%20Syntax%20and%20Semantics%20of%20Programming%20Languages%20-%20Kenneth%20Slonneger.pdf)]

* [[David A. Schmidt]], *Denotational Semantics -- A methodology for language development*, Allyn and Bacon (1986)  [[pdf](https://people.cs.ksu.edu/~schmidt/text/DenSem-full-book.pdf), [webpage](https://people.cs.ksu.edu/~schmidt/text/densem.html)]

* [[Thomas Streicher]], *Domain-Theoretic Foundations of Functional Programming*, World Scientific (2006) [[pdf](https://doc.lagout.org/programmation/Functional%20Programming/Domain-Theoretic%20Foundations%20of%20Functional%20Programming%20%5BStreicher%202006-12-04%5D.pdf), [doi:10.1142/6284](https://www.worldscientific.com/worldscibooks/10.1142/6284)]

See also:

* Wikipedia, [Denotational semantics](http://en.wikipedia.org/wiki/Denotational_semantics)

Discussion of denotational semantics for [[Haskell]]:

* *[Haskell/Denotational semantics](https://en.wikibooks.org/wiki/Haskell/Denotational_semantics)*

category: computer science

[[!redirects denotational semantics]]


dense

* [[dense subspace]]

* [[dense monomorphism]]

* [[nowhere dense set]]

* [[dense topology]]

* [[(dense,closed)-factorization]]

* [[dense subcategory]]

* [[dense subsite]]

* [[dense subtopos]]

* [[dense functor]]

* [[codense functor]]

* [[codensity monad]]

* [[dense subalgebra]]

* [[dense linear order]]

* [[dense relation]]

category: disambiguation