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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{little cubes operad} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{presentation_by_enriched_operads}{Presentation by enriched operads}\dotfill \pageref*{presentation_by_enriched_operads} \linebreak \noindent\hyperlink{as_operads}{As $\infty$-operads}\dotfill \pageref*{as_operads} \linebreak \noindent\hyperlink{Propeties}{Properties}\dotfill \pageref*{Propeties} \linebreak \noindent\hyperlink{GrouplikeMonoid}{Grouplike monoid objects}\dotfill \pageref*{GrouplikeMonoid} \linebreak \noindent\hyperlink{MainResult}{$k$-fold delooping, monoidalness and $\mathbb{E}[k]$-action}\dotfill \pageref*{MainResult} \linebreak \noindent\hyperlink{StabilizationHypothesis}{Stabilization hypothesis}\dotfill \pageref*{StabilizationHypothesis} \linebreak \noindent\hyperlink{AdditivityTheorem}{Additivity theorem}\dotfill \pageref*{AdditivityTheorem} \linebreak \noindent\hyperlink{RelationToFultonMacPhersonOperad}{Relation to Fulton-MacPherson operad}\dotfill \pageref*{RelationToFultonMacPhersonOperad} \linebreak \noindent\hyperlink{cohomology_formality}{Cohomology: Formality}\dotfill \pageref*{cohomology_formality} \linebreak \noindent\hyperlink{homology_poisson_operads}{Homology: Poisson operads}\dotfill \pageref*{homology_poisson_operads} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \textbf{little $k$-disk operad} or \textbf{little $k$-cubes operad} (to distinguish from the [[framed little n-disk operad]]) is the [[topological operad]]/[[(∞,1)-operad]] $E_k$ whose $n$-ary operations are parameterized by rectilinear disjoint embeddings of $n$ $k$-dimensional cubes into another $k$-dimensional cube. When regarded as a [[topological operad]], the topology on the space of all such embedding is such that a continuous path is given by continuously moving the images of these little cubes in the big cube around. Therefore the [[algebra over an operad|algebras over]] the $E_k$ operad are ``$k$-fold monoidal'' objects For instance [[k-tuply monoidal (n,r)-category|k-tuply monoidal (n,r)-categories]]. The limiting [[E-∞ operad]] is a [[resolution]] of the ordinary commutative monoid operad [[Comm]]. Its algebras are [[commutative algebra in an (infinity,1)-category|homotopy commutative monoid objects]] such as $E_\infty$-[[E-infinity-ring|rings]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} (\ldots{}) An [[algebra over an operad]] over $E_k$ is an [[Ek-algebra]]. \hypertarget{presentation_by_enriched_operads}{}\subsubsection*{{Presentation by enriched operads}}\label{presentation_by_enriched_operads} (\ldots{}) \textbf{Remark} Many models for $E_\infty$-operads in the literature are not in fact cofibrant in the [[model structure on operads]], but are $\Sigma$-cofibrant. By the therem at [[model structure on algebras over an operad]], this is sufficient for their categories of algebras to present the correct $\infty$-categories of [[E-∞ algebras]]. (\ldots{}) \hypertarget{as_operads}{}\subsubsection*{{As $\infty$-operads}}\label{as_operads} \begin{defn} \label{}\hypertarget{}{} Fix an integer $k \ge 0$. We let $\square^k = ( -1, 1)^k$ denote an open cube of dimension $k$. We will say that a map $f : \square^k \to \square^k$ is a \emph{rectilinear embedding} if it is given by the formula $f (x_1 , . . . , x_k ) = (a_1 x_1 + b_1 , . . . , a_k x_k + b_k )$ for some real constants $a_i$ and $b_i$ , with $a_i \gt 0$. More generally, if $S$ is a finite set, then we will say that a map $\square^k \times S \to \square^k$ is a rectilinear embedding if it is an open embedding whose restriction to each connected component of $\square^k\times S$ is rectilinear. Let $Rect(\square^k \times S, \square^k )$ denote the collection of all rectitlinear embeddings from $\square^k \times S$ into $\square^k$ . We will regard $Rect(\square^2\times S, \square^k )$ as a topological space (it can be identified with an open subset of $(\mathbf{R}^{2k} )^S )$. The spaces $Rect(\square^k \times \{1, . . . , n\}, \square^k)$ constitute the $n$-ary operations of a topological operad, which we will denote by $tE_k$ and refer to as the \emph{little k-cubes operad}. This is [[Higher Algebra]] Definition 5.1.0.1. \end{defn} \begin{defn} \label{}\hypertarget{}{} We define a topological category $tE^\otimes_k$ as follows: \begin{itemize}% \item The objects of $t E^\otimes_k$ are the objects $[n] \in Fin_*$. \item Given a pair of objects $[m], [n] \in tE^\otimes_k$ , a morphism from $[m]$ to $[n]$ in $t E^\otimes_k$ consists of the following data: \begin{itemize}% \item A morphism $\alpha : [m] \to [n]$ in $Fin_*$ . \item For each $j \in [n]^\circ$ a rectilinear embedding $\square^k \times \alpha^{-1} \{j\} \to \square^k$. \end{itemize} \item For every pair of objects $[m], [n] \in tE^\otimes_k$ , we regard $Hom_{tE^\otimes_k} ([m], [n])$ as endowed with the [[topological space|topology]] induced by the presentation \begin{displaymath} Hom_{tE^\otimes_k} ([m], [n]) = \coprod_{f :\colon [m]\to [n]} \prod_{1\le j\le n} Rect(\times \alpha^{-1} \{j\},\square^k) \,. \end{displaymath} \item Composition of morphisms in $tE^\otimes_k$ is defined in the obvious way. We let $E^\otimes_k$ denote the nerve of the topological category $tE^\otimes_k$. Corollary T.1.1.5.12 implies that $E^\otimes_k$ is an $\infty$-category. There is an evident [[forgetful functor]] from $tE^\otimes_k$ to the (discrete) category $Fin_*$ , which induces a functor $E^\otimes_k \to N(Fin_* )$. \end{itemize} This is [[Higher Algebra]] Definition 5.1.0.2. \end{defn} \hypertarget{Propeties}{}\subsection*{{Properties}}\label{Propeties} \hypertarget{GrouplikeMonoid}{}\subsubsection*{{Grouplike monoid objects}}\label{GrouplikeMonoid} Let $\mathcal{X}$ be an [[(∞,1)-sheaf]] [[(∞,1)-topos]] and $X : Assoc \to \mathcal{X}$ be a monoid object in $\mathcal{X}$. Say that $X$ is \emph{grouplike} if the composite \begin{displaymath} \Delta^{op} \to Ass \to \mathcal{X} \end{displaymath} (see 1.1.13 of [[Commutative Algebra]]) is a [[groupoid object in an (infinity,1)-category|groupoid object]] in $\mathcal{X}$. Say an $\mathbb{E}[1]$-algebra object is grouplike if it is grouplike as an $Ass$-monoid. Say that an $\mathbb{E}[k]$-algebra object in $\mathcal{X}$ is grouplike if the restriction along $\mathbb{E}[1] \hookrightarrow \mathbb{E}[k]$ is. Write \begin{displaymath} Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \subset Mon_{\mathbb{E}[k]}(\mathcal{X}) \end{displaymath} for the [[(∞,1)-category]] of grouplike $\mathbb{E}[k]$-monoid objects. \hypertarget{MainResult}{}\subsubsection*{{$k$-fold delooping, monoidalness and $\mathbb{E}[k]$-action}}\label{MainResult} The following result of (\hyperlink{Lurie}{Lurie}) makes precise for \emph{parameterized [[∞-groupoid]]s} -- for [[∞-stack]]s -- the general statement that $k$-fold [[delooping]] provides a correspondence between [[n-category|n-categories]] that have trivial [[k-morphism|r-morphism]]s for $r \lt k$ and [[k-tuply monoidal n-category|k-tuply monoidal n-categories]]. \begin{utheorem} Let $k \gt 0$, let $\mathcal{X}$ be an [[∞-stack]] [[(∞,1)-topos]] and let $\mathcal{X}_*^{\geq k}$ denote the [[full subcategory]] of the category $\mathcal{X}_{*}$ of pointed objects, spanned by those pointed objects that are $k-1$-[[connected]] (i.e. their first $k$ [[homotopy groups in an (∞,1)-topos|∞-stack homotopy groups]]) vanish. Then there is a canonical equivalence of [[(∞,1)-category|(∞,1)-categories]] \begin{displaymath} \mathcal{X}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \,. \end{displaymath} \end{utheorem} \begin{proof} This is [[Ek-Algebras|EKAlg, theorem 1.3.6.]]. \end{proof} Specifically for $\mathcal{X} = Top$, this refines to the classical theorem by (\hyperlink{May}{May}). \begin{utheorem} Let $Y$ be a [[topological space]] equipped with an action of the [[little cubes operad]] $\mathcal{C}_k$ and suppose that $Y$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$. \end{utheorem} \begin{proof} This is [[Ek-Algebras|EkAlg, theorem 1.3.16.]] \end{proof} Proofs independent of higher order categories can be extracted from the literature. See this \href{http://mathoverflow.net/a/202345/2926}{MO answer} by Tyler Lawson for details. \hypertarget{StabilizationHypothesis}{}\subsubsection*{{Stabilization hypothesis}}\label{StabilizationHypothesis} A proof of the [[stabilization hypothesis]] for [[k-tuply monoidal n-category|k-tuply monoidal n-categories]] is a byproduct of corollary 1.1.10 of (\hyperlink{Lurie}{Lurie}), stated as example 1.2.3. \hypertarget{AdditivityTheorem}{}\subsubsection*{{Additivity theorem}}\label{AdditivityTheorem} It has been long conjectured that it should be true that when suitably defined, there is a tensor product of $\infty$-operads such that \begin{displaymath} \mathbb{E}_k \otimes \mathbb{E}_{k'} \simeq \mathbb{E}_{k + k'} \,. \end{displaymath} This is discussed and realized in section 1.2. of (\hyperlink{Lurie}{Lurie}). The tensor product is defined in appendix B.7. \hypertarget{RelationToFultonMacPhersonOperad}{}\subsubsection*{{Relation to Fulton-MacPherson operad}}\label{RelationToFultonMacPhersonOperad} \begin{prop} \label{}\hypertarget{}{} The [[Fulton-MacPherson operad]] is [[weak equivalence|weakly equivalent]] in the [[model structure on operads]] with respect to the [[classical model structure on topological spaces]], to the [[little n-disk operad]] \end{prop} (\hyperlink{Salvatore01}{Salvatore 01, Prop. 4.9}, summarized as \hyperlink{LambrechtsVolic14}{Lambrechts-Volic 14, Prop. 5.6}) \hypertarget{cohomology_formality}{}\subsubsection*{{Cohomology: Formality}}\label{cohomology_formality} [[the little n-disk operad is formal]] \hypertarget{homology_poisson_operads}{}\subsubsection*{{Homology: Poisson operads}}\label{homology_poisson_operads} For an $E_k$-operad in a [[category of chain complexes]], its [[homology]] is the [[Poisson operad]] $P_{k}$. See for instance (\hyperlink{Costello}{Costello}) and see at \emph{[[Poisson n-algebra]]}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} Explicit models of $E_\infty$-operads include \begin{itemize}% \item [[Barratt-Eccles operad]] \end{itemize} (\ldots{}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[configuration space of points]], [[Fulton-MacPherson operad]] \item [[factorization algebra]] \item [[A-∞ operad]] \item [[E-∞ operad]] \item [[L-∞ operad]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Review includes \begin{itemize}% \item [[Benoit Fresse]], \emph{Little discs operads, graph complexes and Grothendieck--Teichmüller groups}, in [[Haynes Miller]] (ed.) \emph{[[Handbook of Homotopy Theory]]} (\href{https://arxiv.org/abs/1811.12536}{arXiv:1811.12536}) \end{itemize} A standard textbook reference is chapter 4 of \begin{itemize}% \item [[Peter May]], \emph{The geometry of iterated loop spaces} (\href{http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf}{pdf}) \end{itemize} The equivalence to the [[Fulton-MacPherson operad]] is due to \begin{itemize}% \item [[Paolo Salvatore]], \emph{Configuration spaces with summable labels}, Cohomological methods in homotopy theory. Birkhäuser, Basel, 2001. 375-395. \end{itemize} Proof that [[the little n-disk operad is formal]] was sketched by [[Maxim Kontsevich]] and spelled out in \begin{itemize}% \item [[Pascal Lambrechts]], [[Ismar Volić]], section 5 of \emph{Formality of the little N-disks operad}, Memoirs of the American Mathematical Society ; no. 1079, 2014 (\href{http://dx.doi.org/10.1090/memo/1079}{doi:10.1090/memo/1079}) \end{itemize} [[John Francis]]` work on $E_n$-actions on $(\infty,1)$-categories is in \begin{itemize}% \item [[John Francis]] PhD thesis (\href{http://dspace.mit.edu/handle/1721.1/43792}{web}) \end{itemize} This influenced the revised version of \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Commutative Algebra]]} (\href{http://arxiv.org/abs/math/0703204}{arXiv}) \end{itemize} and is extended to include a discussion of traces and centers in \begin{itemize}% \item [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], \emph{Integral transforms and Drinfeld Centers in Derived Geometry} (\href{http://arxiv.org/abs/0805.0157}{arXiv}) (see also [[geometric ∞-function theory]]) \end{itemize} A detailed discussion of $E_k$ in the context of [[(∞,1)-operads]] is in \begin{itemize}% \item [[Jacob Lurie]], \emph{$\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf}{pdf}) \end{itemize} An elementary computation of the [[homology]] of the little $n$-disk operad in terms of \emph{solar system calculus} is in \begin{itemize}% \item [[Dev Sinha]], \emph{The homology of the little disks operad} (\href{http://arxiv.org/abs/math/0610236}{arXiv:math/0610236}) \end{itemize} For the relation to Poisson Operads see \begin{itemize}% \item [[Kevin Costello]], [[Owen Gwilliam]], \emph{Factorization algebras in perturbative quantum field theory : $P_0$-operad} \end{itemize} [[!redirects little k-cubes operad]] [[!redirects little n-cubes operad]] [[!redirects little cube operad]] [[!redirects little k-cube operad]] [[!redirects little n-cube operad]] [[!redirects little discs operad]] [[!redirects little disc operad]] [[!redirects little k-discs operad]] [[!redirects little n-discs operad]] [[!redirects little k-disc operad]] [[!redirects little n-disc operad]] [[!redirects little disks operad]] [[!redirects little disk operad]] [[!redirects little k-disks operad]] [[!redirects little n-disks operad]] [[!redirects little k-disk operad]] [[!redirects little n-disk operad]] [[!redirects little n-disk operad]] [[!redirects little n-disk operads]] [[!redirects Ek-operad]] [[!redirects Ek-operads]] [[!redirects E-k operad]] [[!redirects E-n operad]] [[!redirects E-k-operad]] [[!redirects E-n-operad]] [[!redirects E-k operads]] [[!redirects E-n operads]] [[!redirects E-k-operads]] [[!redirects E-n-operads]] [[!redirects En operad]] [[!redirects En-operad]] [[!redirects En-operads]] [[!redirects E-k operad]] [[!redirects E-k operad]] [[!redirects little disk operad]] [[!redirects little 2-disk operad]] \end{document}