Smooth $A_{\infty}$-form on a diffeological loop space

A site is a concrete category with a ‘coverage’ assigning a ‘covering family’ to each object. For a site ${\sf{C}}$, we denote by $\operatorname{Obj}\,(\text{\small${\sf{C}}$})$ the class of objects, by $\operatorname{Mor}_{\text{\small\,${\sf{C}}$\,}}(A,B)$ the set of morphisms from $A$ to $B$, and by $\operatorname{Cov}_{\text{\small\,${\sf{C}}$\,}}(U)$ the set of covering families on $U\in\operatorname{Obj}\,(\text{\small${\sf{C}}$})$. We denote by ${\sf{Set}}$ the category of sets and maps between sets. For a given set $X$, we have two contravariant functors $\mathcal{M}_{X},\,\mathcal{K}_{X}:{\sf{C}}\to{\sf{Set}}$ defined by

1. (1)

   $\mathcal{M}_{X}(U)=\operatorname{Map}(U,X)$ the set of maps from $U$ to $X$ and

2. (2)

   $\mathcal{K}_{X}(U)=\{\,P\in\mathcal{M}_{X}(U)\,\,\mathstrut\vrule\,\,\text{$P:U\to X$ is locally constant}\,\}$,

where we say $P:U\to X$ is locally constant, if there exists a covering family $\{\,g_{\alpha}:V_{\alpha}\to U\,\}_{\alpha\in\Lambda}$ of $U$ such that $P{\smash{\lower-0.43057pt\hbox{\scriptsize$\circ$\,}}}g_{\alpha}$ is constant for any $\alpha\in\Lambda$.

In [Che73, Che75, Che77, Che86], K. T. Chen introduced a site ${\sf{{Convex}}}$ which is a category of convex sets with non-void interiors in $\mathbb{R}^{n}$ for some $n\!\geq\!0$, and smooth functions between them in the ordinary sense (see [KM97]), with a ‘coverage’ assigning a ‘covering family’ to each convex set with non-void interior, which is the set of open coverings by interiors of convex sets.

In [Sou80], J. M. Souriau introduced a similar but a slightly more sophisticated site ${{\sf{Domain}}}$ which is a category of open sets in $\mathbb{R}^{n}$ for some $n\!\geq\!0$, and smooth functions between them in the ordinary sense, with a ‘coverage’ assigning a ‘covering family’ to each open set, which is the set of open coverings in the usual sense.

We call a pair $(X,\mathcal{D}_{X})$ a diffeological space, if it satisfies the following conditions.

1. (D1)

   $X$ is a set and $\mathcal{D}_{X}:{{\sf{Domain}}}\to{\sf{Set}}$ is a contravariant functor.

2. (D2)

   For any $U\in\operatorname{Obj}\,(\text{\small${{\sf{Domain}}}$})$, $\mathcal{K}_{X}(U)$ $\subset$ $\mathcal{D}_{X}(U)$ $\subset$ $\mathcal{M}_{X}(U)$.

3. (D3)

   For any $U\!\in\!\operatorname{Obj}\,(\text{\small${{\sf{Domain}}}$})$ and any $P\in\mathcal{M}_{X}(U)$, $P\in\mathcal{D}_{X}(U)$ if there exists $\{U_{\alpha}\}_{\alpha\in\Lambda}\in\operatorname{Cov}_{\text{\small\,${{\sf{Domain}}}$\,}}(U)$ such that $P|_{U_{\alpha}}\in\mathcal{D}_{X}(U_{\alpha})$ for all $\alpha\in\Lambda$.

A map $f:X\to Y$ is said to be smooth, if the natural transformation $f_{\!\hbox{\footnotesize$*$}}:\mathcal{M}_{X}\to\mathcal{M}_{Y}$ satisfies $f_{\!\hbox{\footnotesize$*$}}(\mathcal{D}_{X}(U))\subset\mathcal{D}_{Y}(U)$ for any $U\!\in\!\operatorname{Obj}\,(\text{\small${{\sf{Domain}}}$})$. We denote by ${\sf{Diffeology}}$, the category of diffeological spaces and smooth maps between diffeological spaces. An element of $\mathcal{D}_{X}(U)$ is called a plot of $X$ on $U$, and $\mathcal{D}=\bigcup_{U\in\operatorname{Obj}\,(\text{\small${{\sf{Domain}}}$})}\mathcal{D}_{X}(U)$ is called a ‘diffeology’ on $X$. If we replace the site ${{\sf{Domain}}}$ by the site ${\sf{{Convex}}}$, we obtain Chen’s smooth category denoted by ${\sf{Chen}}$. From now on, we discuss in the smooth category ${\sf{Diffeology}}{}$, rather than ${\sf{Chen}}{}$, while we believe that entirely similar arguments can be performed also in ${\sf{Chen}}{}$. Let $\mathbb{N}$ be the set of non-negative integers.

In this paper, a manifold is assumed to be paracompact. We denote by ${\sf{Manifold}}$ the category of smooth manifolds and smooth maps between them which can be embedded into ${\sf{Diffeology}}{}$ as a full subcategory (see [IZ13]). One of the advantage to expand our playground to ${\sf{Diffeology}}{}$ than to restrict ourselves in ${\sf{Manifold}}$ is that the category ${\sf{Diffeology}}{}$ is cartesian-closed, complete and cocomplete (see [IZ13]).

The path space in ${\sf{Diffeology}}{}$ is defined using the real line $\mathbb{R}$ in place of the closed interval $[0,1]$ (see [IZ13, Chapter 5]). This definition gives a nice diffeology on a path space, while it causes a technical issue on concatenation:

A work-around can easily be found as in [IZ13, art.5.4] by compressing the moving part into an open subinterval $(\varepsilon,1{-}\varepsilon)\subset(0,1)\subset\mathbb{R}$, where $0<\varepsilon\ll 1$:

On the other hand, if we consider $A_{\infty}$-form of concatenations using $\operatorname{stPaths}_{\varepsilon}(X)$, we need some more tricks to concatenate many paths. In this paper, we adopt slightly different ways to consider a smooth (h-unital) $A_{\infty}$-form for a concatenation.

Let ${\operatorname{\mathcal{P}}(X)}=\{\,u\in\mid u{\smash{\lower-0.43057pt\hbox{\scriptsize$\circ$\,}}}\pi_{set}=u\,\}$, where $\pi_{set}:\mathbb{R}\to\mathbb{R}$ is a continuous idempotent, i.e, $\pi_{set}{\smash{\lower-0.43057pt\hbox{\scriptsize$\circ$\,}}}\pi_{set}=\pi_{set}$, which is defined as follows:

Then by definition, $\pi_{set}$ enjoys the following properties. (1)  $\pi_{set}(t)=0$, $t\leq 0$,     (2)  $\pi_{set}(t)+\pi_{set}(1{-}t)=1$,     (3)  $\pi_{set}(t)=t$, $0<t<1$.

Let us recall basic properties on subductions in Diffeology used in this paper.

From now on, we assume that $L$, $Y$ and $X$ are diffeological spaces, and that $\varpi:L\to Y$ is a subduction. Here, we remark that $\dim{Y}\leq\dim{L}$ diffeologically.

We further assume that $Y$ is a diffeological quotient $L/\varpi_{set}$ by a relation $\varpi_{set}$ on $L$, i.e, $Y=\{\,y\mid\exists\,{t\in L}\ y=\widehat{\varpi}_{set}(t)\,\}$ and $\varpi(t)=\widehat{\varpi}_{set}(t)$, where $\widehat{\varpi}_{set}$ is an equivalence relation on $L$ generated by $\varpi_{set}$. Hence $\varpi{\smash{\lower-0.43057pt\hbox{\scriptsize$\circ$\,}}}\varpi_{set}=\varpi$ as relations from $L$ to $Y$.

If the relation $\varpi_{set}$ is a continuous idempotent on $L$, then $Y$ is topologically the same as $\operatorname{Im}\varpi_{set}\subset L$, while $Y$ might not be a diffeological subspace of $L$.

In Topology, we use the symbol $\mathbb{I}_{top}$ for the topological subspace $[0,1]$ of $\mathbb{R}$. We remark that the topology of $\mathbb{I}_{top}=[0,1]$ is the same as the quotient topology induced by a continuous map $\pi_{top}:\mathbb{R}\to\mathbb{I}_{top}$ given by $\pi_{top}(t)=\pi_{set}(t)\in[0,1]$.

Now we introduce a generalised notion of a simplicial or cubical complex using an idea of a cubic set: a $q$-cubic set $\sigma$ in $\mathbb{R}^{n}$ is defined as a convex body in some affine subspace $L_{\sigma}$ in $\mathbb{R}^{n}$, inductively on $q$, $-1\!\leq\!q\!\leq\!n$ (see also [II19]).

1. (1)

   The $-1$-cubic set in $\mathbb{R}^{n}$ is the empty set $\emptyset\subset\mathbb{R}^{n}$. In this case, $L_{\emptyset}=\emptyset$.

2. (2)

   A $0$-cubic set in $\mathbb{R}^{n}$ is a point $\mathbb{p}\in\mathbb{R}^{n}$. In this case, $L_{\mathbb{p}}=\{\mathbb{p}\}$.

3. (3)

   Let $\sigma_{1}$ and $\sigma_{2}$ be respectively $q_{1}$-cubic and $q_{2}$-cubic sets in $\mathbb{R}^{n}$ with $-1\leq q{-}1\leq q_{1}{+}q_{2}\leq q\leq n$, where $\sigma_{1}$ and $\sigma_{2}$ are convex bodies in affine subspaces $L_{1}$ and $L_{2}$, respectively. Let $V_{1}$ and $V_{2}$ be vector subspaces of $\mathbb{R}^{n}$ such that $V_{1}\cap V_{2}=\{\mathbb{0}\}$, $L_{1}=\mathbb{a}_{1}\!+\!V_{1}$ and $L_{2}=\mathbb{a}_{2}\!+\!V_{2}$ for some $\mathbb{a}_{1}\in\sigma_{1}$ and $\mathbb{a}_{2}\in\sigma_{2}$.

   1. ($q_{1}\!+\!q_{2}=q{-}1$ and $L_{1}\!\cap\!L_{2}=\emptyset$ (or $\mathbb{a}_{2}\!-\!\mathbb{a}_{1}\not\in V_{1}\!+\!V_{2}$))

      The subset $\sigma_{1}\,\hbox{\footnotesize$*$}\,\sigma_{2}$ $=$ $\{\,(1{-}t){\cdot}\mathbb{x}$ ${+}$ $t{\cdot}\mathbb{y}\,;\,\mathbb{x}\!\in\!\sigma_{1},\,\mathbb{y}\!\in\!\sigma_{2},\,t\!\in\!I_{top}\,\}\subset\mathbb{R}^{n}$ is a $q$-cubic set in $\mathbb{R}^{n}$. In this case, we have a relative homeomorphism $\phi_{\sigma_{1},\sigma_{2}}:(\sigma_{1}\times I_{top}\times\sigma_{2},\sigma_{1}\times\{0,1\}\times\sigma_{2})\to(\sigma_{1}\,\hbox{\footnotesize$*$}\,\sigma_{2},\sigma_{1}\amalg\sigma_{2})$ given by $\phi_{\sigma_{1},\sigma_{2}}(\mathbb{x},t,\mathbb{y})=(1{-}t){\cdot}\mathbb{x}\!+\!t{\cdot}\mathbb{y}$.

   2. ($q_{1}\!+\!q_{2}\!=\!q$ and $L_{1}\!\cap L_{2}\not=\emptyset$ (or $\mathbb{a}_{2}\!-\!\mathbb{a}_{1}\in V_{1}\!+\!V_{2}$))

      Let $L_{1}\!\cap L_{2}\!=\!\{\mathbb{a}\}$, $\mathbb{a}\!\in\!\mathbb{R}^{n}$. Then the subset $\sigma_{1}{\,{}_{L_{1}\!\!}\times_{\,L_{2}}}\sigma_{2}=\{\mathbb{x}\,{+}\,\mathbb{y}\,{-}\,\mathbb{a}\,;\,\mathbb{x}\!\in\!\sigma_{1},\,\mathbb{y}\!\in\!\sigma_{2}\}$ is a $q$-cubic set in $\mathbb{R}^{n}$. In this case, we have a homeomorphism $\psi_{\sigma_{1},\sigma_{2}}:\sigma_{1}\times\sigma_{2}\to\sigma_{1}\,{}_{L_{1}\!\!}\times_{\,L_{2}}\sigma_{2}$ given by $\psi_{\sigma_{1},\sigma_{2}}(\mathbb{x},\mathbb{y})=\mathbb{x}\!+\!\mathbb{y}\!-\!\mathbb{a}$.

For each $n\geq 0$ and $q$ with $-1\leq q\leq n$, we denote by $C(n)^{q}$ the set of all $q$-cubic sets in $\mathbb{R}^{n}$ and $C(n)=\{\emptyset\}\cup\underset{q\geq 0}{\cup}C(n)^{q}$. Then the above construction yields two natural products: the join $\hbox{\footnotesize$*$}:C(n)^{q}\times C(n^{\prime})^{q^{\prime}}\to C(n{+}n^{\prime}{+}1)^{q+q^{\prime}+1}$ induced by (3) above using $\mathbb{R}^{n}\approx\mathbb{R}^{n}\times\{0\}\times\{\mathbb{0}\}=V_{1}\subset\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n^{\prime}}\supset V_{2}=\{\mathbb{0}\}\times\{0\}\times\mathbb{R}^{n^{\prime}}\approx\mathbb{R}^{n^{\prime}}$ with $\mathbb{a}_{t}=(\mathbb{0},t,\mathbb{0})$ for $t=1,2$, and the product $\times:C(n)^{q}\times C(n^{\prime})^{q^{\prime}}\to C(n{+}n^{\prime})^{q+q^{\prime}}$ induced by (3) above using $\mathbb{R}^{n}\approx\mathbb{R}^{n}\times\{\mathbb{0}\}=V_{1}\subset\mathbb{R}^{n}\times\mathbb{R}^{n^{\prime}}\supset V_{2}=\{\mathbb{0}\}\times\mathbb{R}^{n^{\prime}}\approx\mathbb{R}^{n^{\prime}}$ with $\mathbb{a}=\mathbb{a}_{1}=\mathbb{a}_{2}=(\mathbb{0},\mathbb{0})$.

The notion of a face of a cubic set is inductively given as follows.

1. (1)

   Let $\sigma$ be a cubic set. Then the emptyset $\emptyset$ and $\sigma$ itself are faces of $\sigma$.

2. (2)

   Let $\sigma_{1}$ and $\sigma_{2}$ be two cubic set. Then we have the following.

   1. (a)

      A face of $\sigma_{1}\,\hbox{\footnotesize$*$}\,\sigma_{2}$ is expressed as $\tau_{1}\,\hbox{\footnotesize$*$}\,\tau_{2}$ for some faces $\tau_{1}$ and $\tau_{2}$ of $\sigma_{1}$ and $\sigma_{2}$, respectively. Therefore $\sigma_{1}=\sigma_{1}\,\hbox{\footnotesize$*$}\,\emptyset$ and $\sigma_{2}=\emptyset\,\hbox{\footnotesize$*$}\,\sigma_{2}$ are faces of $\sigma_{1}\,\hbox{\footnotesize$*$}\,\sigma_{2}$.

   2. (b)

      A face of $\sigma_{1}\,{}_{L_{1}\!\!}\times_{\,L_{2}}\sigma_{2}$ is expressed as $\tau_{1}\,{}_{L_{1}\!\!}\times_{\,L_{2}}\tau_{2}$ for some faces $\tau_{1}$ and $\tau_{2}$ of $\sigma_{1}$ and $\sigma_{2}$, respectively.

We denote $\tau\prec\sigma$ if $\tau\in C(n)$ is a face of $\sigma\in C(n)$.

An ordered subset $\mathbb{K}\subset C(n)$ is called a cubic complex, if the following holds. (0)  $\forall\,\tau,\,\sigma\in\mathbb{K}\ \ \tau\cap\sigma\in C(n),\ \tau\cap\sigma\prec\tau\ \text{and}\ \tau\cap\sigma\prec\sigma$.     (1)  $\emptyset\in\mathbb{K}$,     (2)  $\forall\,\tau\in C(n)\ \,\forall\,\sigma\in\mathbb{K}\ \ \tau\prec\sigma\implies\tau\in\mathbb{K}$, A subset $\mathbb{L}\subset\mathbb{K}$ with the following properties is called a cubic subcomplex of $\mathbb{K}$. (1)  $\emptyset\in\mathbb{L}$,     (2)  $\forall\,\tau\in\mathbb{K}\ \,\forall\,\sigma\in\mathbb{L}\ \ \tau\prec\sigma\implies\tau\in\mathbb{L}$. Then we denote $\dim{\mathbb{K}}=\max\{\,\dim\sigma\mid\sigma\in\mathbb{K}\,\}$, where $\dim\sigma=q$ if $\sigma\in C(n)^{q}$.

For any $q$-cubic set $\sigma\in K$, $\mathbb{K}(\sigma)=\{\,\tau\!\in\!C(n)\,\,\mathstrut\vrule\,\,\tau\preceqq\sigma\,\}$ for $q\geq-1$ and $\mathbb{K}(\dot{\sigma})=\{\tau\!\in\!C(n)\,\,\mathstrut\vrule\,\,\tau\precneqq\sigma\}$ for $q\geq 0$ are cubic subcomplexes of $\mathbb{K}$.

For any two cubic complexes $\mathbb{K}\subset C(n)$ and $\mathbb{L}\subset C(m)$, we obtain

1. (1)

   $\mathbb{K}\,\hbox{\footnotesize$*$}\,\mathbb{L}:=\{\,\sigma\,\hbox{\footnotesize$*$}\,\tau\,\,\mathstrut\vrule\,\,\sigma\!\in\!\mathbb{K},\tau\!\in\!\mathbb{L}\,\}\subset C(n{+}m{+}1)$

2. (2)

   $\mathbb{K}\times\mathbb{L}:=\{\,\sigma\times\tau\,\,\mathstrut\vrule\,\,\sigma\!\in\!\mathbb{K},\tau\!\in\!\mathbb{L}\,\}\subset C(n{+}m)$

For any cubic complexes $\mathbb{K}\subset C(n)$ and $\mathbb{K}^{\prime}\subset C(m)$, an order-preserving map $\varphi:\mathbb{K}\rightarrow\mathbb{K}^{\prime}$ is called a cubic map, if the following conditions are satisfied. (1)  $\varphi^{-1}(\emptyset)=\{\emptyset\}$,     (2)  $\forall\,\tau^{\prime}\in\mathbb{K}^{\prime}\ \,\forall\,\sigma\in\mathbb{K}\ \ \tau^{\prime}\prec\varphi(\sigma)\implies\exists\,\tau\prec\sigma$$\varphi(\tau)=\tau^{\prime}$. In particular, the image of a cubic map $\varphi:\mathbb{K}\rightarrow\mathbb{K}^{\prime}$ is a cubic subcomplex of $\mathbb{K}^{\prime}$.

For a cubic complex $\mathbb{K}\!\subset\!C(n)$, $n\!\geq\!0$, we denote $\mathbb{K}^{q}=\{\,\sigma\!\in\!K\,;\,\text{$\sigma$ is $q$-{cubic}}\,\}$, $q\!\geq\!-1$ and by $|{\mathbb{K}}|=\underset{\sigma\in\mathbb{K}}{\bigcup}\,\sigma$ the polyhedron in $\mathbb{R}^{n}$ associated to $\mathbb{K}$. For a cubic complexes $\mathbb{K}$ and $\mathbb{L}$, a continuous map $f:|{\mathbb{L}}|\to|{\mathbb{K}}|$ is called cubic, if there exists a map $\varphi:\mathbb{L}\to\mathbb{K}$ such that $f|_{\tau}:\tau\to\varphi(\tau)\subset|{\mathbb{K}}|$ for any $\tau\in\mathbb{L}$. Such a map $f$ is often denoted by $|\varphi|:|{\mathbb{L}}|\to|{\mathbb{K}}|$.

In Diffeology, we use the symbol $\mathbb{I}$ for a special diffeological space: let $\mathbb{I}=\mathbb{R}/{\pi_{set}}$ be the diffeological quotient of $\mathbb{R}$, where $\mathbb{I}=[0,1]$ as a set, with a subduction $\pi:\mathbb{R}\to\mathbb{I}$ given by $\pi(t)=\pi_{set}(t)\in[0,1]$, so that we obtain $\dim\mathbb{I}=1$ diffeologically and $\pi{\smash{\lower-0.43057pt\hbox{\scriptsize$\circ$\,}}}\pi_{set}=\pi$. The underlying topology ($D$-Topology in [IZ13, 2.8]) of $\mathbb{I}$ is the same as $\mathbb{I}_{top}\subset\mathbb{R}$, while $\mathbb{I}\hookrightarrow\mathbb{R}$ can not be an induction.

As is well-known, there is a smooth function $\lambda:\mathbb{R}\to\mathbb{R}$ enjoying (1)  $\lambda(t)=0$, $t\leq 0$,     (2)  $\lambda(t)+\lambda(1{-}t)=1$,     (3)  $\lambda^{\prime}(t)>0$, $0<t<1$.

For any $q$-cubic set $\sigma$, we clearly have $\dim\sigma=q$ diffeologically, and by Proposition 1.3, we obtain that $\pi_{\sigma}^{*}:C^{\infty}(\sigma,X)\rightarrow C^{\infty}(\mathbb{R}^{q},X)$ is an induction.

For a cubic complex $\mathbb{K}\subset C(n)$, we introduce a smooth structure on the polyhedron $|\mathbb{K}|$ as $|{\mathbb{K}}|=\underset{\sigma\in\mathbb{K}}{\operatorname{colim}}\,\sigma$, which is called a smooth cubic polyhedron. Then by definition, we obtain a smooth injection $\hat{\lambda}_{\mathbb{K}}:|\mathbb{K}|\to\mathbb{R}^{n}$ by collecting smooth maps $\hat{\lambda}_{\sigma}:\sigma\to\mathbb{R}^{n}$, $\sigma\in\mathbb{K}$. Then we instantly see that $\dim{|\mathbb{K}|}=\dim{\mathbb{K}}$ diffeologically.

Let us introduce associahedra $K_{n}\subset\mathbb{R}^{n}$, $n\geq 1$ in ${\sf{Topology}}{}$ as follows, which is slightly modified from the definition by Stasheff (see [Sta63], [IM89] or [Iwa12]):

or equivalently, we can describe the associahedron as follows.

Let $H_{1}^{n-2}$​ : ​$x_{1}+{\cdots}+x_{n}\!=\!n{-}1,x_{1}\!=\!0$ be an affine space where $K_{n}$ is a convex body.

Let $A(n)=\{\,(k,r,s)\in\mathbb{N}\,\,\mathstrut\vrule\,\,1\!\leq\!k\!\leq\!r,\ 2\!\leq\!s\!=\!n{-}r{+}1\!\leq\!n{-}1\,\}$. Then the boundary of $K_{n}$ is the union of faces corresponding to elements in $A(n)$, given as follows.