\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\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*{H-space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{loop_spaces}{Loop spaces}\dotfill \pageref*{loop_spaces} \linebreak
\noindent\hyperlink{suspensions}{Suspensions}\dotfill \pageref*{suspensions} \linebreak
\noindent\hyperlink{spheres}{Spheres}\dotfill \pageref*{spheres} \linebreak
\noindent\hyperlink{mapping_spaces_into_hgroups}{Mapping spaces into H-groups}\dotfill \pageref*{mapping_spaces_into_hgroups} \linebreak
\noindent\hyperlink{ktheory_space}{K-Theory space}\dotfill \pageref*{ktheory_space} \linebreak
\noindent\hyperlink{dwyerwilkerson_space}{Dwyer-Wilkerson space}\dotfill \pageref*{dwyerwilkerson_space} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_spaces}{Relation to $A_\infty$-spaces}\dotfill \pageref*{relation_to_spaces} \linebreak
\noindent\hyperlink{RingStructureOnHomology}{Ring structure on homology}\dotfill \pageref*{RingStructureOnHomology} \linebreak
\noindent\hyperlink{hspaces_in_homotopy_type_theory}{H-Spaces in Homotopy Type theory}\dotfill \pageref*{hspaces_in_homotopy_type_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An \textbf{$H$-space} (``H'' for [[Heinz Hopf|Hopf]], as in \emph{[[Hopf construction]]}) is a [[magma]] [[internalization|internal to]] the [[classical homotopy category]] of [[topological spaces]] [[Ho(Top)]], or in the homotopy category category $Ho(Top)_*$ of [[pointed topological spaces]], which has a [[unit]] up to [[homotopy]].

Similarly:

An \textbf{$H$-monoid} is a [[monoid object]] in [[Ho(Top)]], hence an $H$-space is an $H$-monoid if the product of the magma is [[associativity|associative]] up to [[homotopy]].

An \textbf{$H$-group} is a [[group object]] in [[Ho(Top)]], so an $H$-monoid is an $H$-group if it also has [[inverses]] up to homotopy.

An \textbf{$H$-ring} is a [[ring object]] in (pointed) [[Ho(Top)]].

To continue in this pattern, one could say that an \textbf{H-category} is an [[Ho(Top)]]-[[enriched category]].

Notice that here the homotopies for units, associativity etc. are only required to \emph{exist} for an H-space, not required to be equipped with higher [[coherence|coherent]] homotopies. An $H$-monoid equipped with such higher and coherent homotopies is instead called a \emph{strongly homotopy associative} space or \emph{$A_\infty$-[[A-infinity space|space]]} for short. If it has only higher homotopies up to level $n$, it is called an $A_n$-[[A-n space|space]].

A better name for an $H$-space might be be $H$-[[unitoid]], but it is rarely used. The $H$ stands for [[Heinz Hopf]], and reflects the sad fact that the natural name `homotopy group' was [[homotopy group|already occupied]]; Hopf and A. Borel found necessary algebraic conditions for a space to admit an $H$-space structure.

There are dual notions of $H$-counitoid (or $H'$-space, or [[co-H-space]]), $H$-comonoid (or $H'$-monoid) and $H$-[[cogroup]] (or $H'$-group) having co-operations with the usual identities up to homotopy.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{loop_spaces}{}\subsubsection*{{Loop spaces}}\label{loop_spaces}

The main example of an $H$-group is the [[loop space]] $\Omega X$ of a space $X$, which however is naturally even an [[A-infinity space]].

\hypertarget{suspensions}{}\subsubsection*{{Suspensions}}\label{suspensions}

The main example of an [[co-H-space|H-cogroup]] in $Top_*$ is the [[suspension]] $S X= S^1\wedge X$ of a pointed topological space $X$.

\hypertarget{spheres}{}\subsubsection*{{Spheres}}\label{spheres}

The only [[n-spheres]] $S^n$ which have $H$-space structure are those for $n = 0,1,3,7$, and their H-space structure is given by identifying them as the unit spheres in one of the four [[normed division algebras]] ([[real numbers]], [[complex numbers]], [[quaternions]], [[octonions]]) and taking the product to be that induced by the algebra product.

An example of an H-space that does not lift to an [[A-infinity space]] is the [[7-sphere]] $S^7$. It can't be delooped because its [[delooping]] would have [[cohomology group]] a [[polynomial ring]] on a generator in degree 8, and this is impossible by mod $p$ [[Steenrod operations]] for any odd $p$, see \hyperlink{Adams61}{Lemma 2, Adams61}. The 7-sphere is also not an $H$-group.

\hypertarget{mapping_spaces_into_hgroups}{}\subsubsection*{{Mapping spaces into H-groups}}\label{mapping_spaces_into_hgroups}

If $K$ is an $H$-group then for any topological space $X$, the set of [[homotopy classes]] $[X,K]$ has a natural group structure in the strict sense; analogously if $K'$ is an $H$-cogroup then $[K',X]$ has a group structure. If there is more than one $H$-group structure on a space, then the induced group structures on the set of homotopy classes coincide.

If an $H$-space is equivalent to a [[delooping|deloopable one]], then it is a [[groupoid object in an (infinity,1)-category|group object in the (∞,1)-category]] [[Top]]. In other words, in that case, the associativity and other axioms hold up to \textbf{coherent homotopy}.

\hypertarget{ktheory_space}{}\subsubsection*{{K-Theory space}}\label{ktheory_space}

The [[classifying space]] $B U \times \mathbb{Z}$ for (complex) [[topological K-theory]] is an H-ring space (p. 205 (213 of 251) in \emph{[[A Concise Course in Algebraic Topology]]}.)

\hypertarget{dwyerwilkerson_space}{}\subsubsection*{{Dwyer-Wilkerson space}}\label{dwyerwilkerson_space}

See at \emph{[[Dwyer-Wilkerson H-space]]}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_spaces}{}\subsubsection*{{Relation to $A_\infty$-spaces}}\label{relation_to_spaces}

Given an [[A-∞ space]] in the [[(∞,1)-category]] [[∞Grpd]]/[[Top]], its image in the corresponding [[homotopy category of an (∞,1)-category]] [[Ho(Top)]] in an $H$-space. Conversely, refining an H-space to a genuine $A_\infty$-space means lifting the structure of a [[monoid object in an (∞,1)-category]] from the [[homotopy category of an (∞,1)-category|homotopy category]] [[Ho(Top)]] to the genuine [[(∞,1)-category]] [[∞Grpd]]/[[Top]].

Further discussion of this is also at \emph{\href{loop+space#AInfinityStructure}{loop space -- Homotopy associative structure}}.

\hypertarget{RingStructureOnHomology}{}\subsubsection*{{Ring structure on homology}}\label{RingStructureOnHomology}

The operations on an [[H-space]] $X$ equip its [[homology]] with the [[mathematical structure|structure]] of [[ring]]. At least for [[ordinary homology]] this is known as the \emph{[[Pontrjagin ring]]} $H_*(X)$ of $X$.

\hypertarget{hspaces_in_homotopy_type_theory}{}\subsection*{{H-Spaces in Homotopy Type theory}}\label{hspaces_in_homotopy_type_theory}

See: [[homotopytypetheory:H-space]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[topological monoid]]


\item [[Pontrjagin ring]]


\item [[group completion]]


\item [[H-space ring spectrum]], [[H-group ring spectrum]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Introduction and survey includes

\begin{itemize}%
\item [[Martin Arkowitz]], \emph{H-Spaces and Co-H-Spaces}, chapter 2 in \emph{Introduction to homotopy theory}, Springer 2011 (\href{file:///C:/Users/Sony/Downloads/9781441973283-c1.pdf}{pdf})

\end{itemize}
The terminology $H$-space is a definition in a Chapter IV, Section 1 (dedicated to loop spaces) of

\begin{itemize}%
\item [[Jean-Pierre Serre]], \emph{Homologie singuli\`e{}re des espaces fibr\'e{}s. Applications.}, Ann. Math. \textbf{54} (1951), 425--505.

\end{itemize}
Some other papers in the 1950s include

\begin{itemize}%
\item [[Edwin Spanier]], [[Henry Whitehead]], \emph{On fibre spaces in which the fibre is contractible}, Comment. Math. Helv. \textbf{29}, 1955, 1--8.


\item Arthur H. Copeland, \emph{On $H$-spaces with two nontrivial homotopy groups}, Proc. AMS \textbf{8}, 1957, 109--129.


\item M. Sugawara:

\begin{itemize}%
\item \emph{$H$-spaces and spaces of loops}, Math. J. Okayama Univ, \textbf{5}, 1955, 5--11;
\item \emph{A condition that a space is an $H$-space}, Math. J. Okayama Univ. \textbf{6}, 1957, 109--129;
\item \emph{A condition that a space is group-like}, Math. J. Okayama Univ. \textbf{7}, 1957, 123--149.

\end{itemize}

\item F. I. Karpelevi, A. L. Oni\v{s}ik, \emph{Algebra of homologies of space of paths}, Dokl. Akad. Nauk. SSSR, (N.S.), 106 (1956), 967--969. MR0081478



\end{itemize}
The theory of $H$-spaces was widely established in the 1950s and studied by Serre, Postnikov, Spanier, Whitehead, Dold, Eckmann and Hilton and many others.

The deloopable case has more coherent structure which has been discovered few years later in J. Stasheff's thesis and published in

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Homotopy associative $H$-spaces I, II}, Trans. Amer. Math. Soc. 108, 1963, 275-312

\end{itemize}
For a historical account see

\begin{itemize}%
\item [[John McCleary]], \emph{An appreciation of the work of Jim Stasheff} (\href{http://www.math.unc.edu/Faculty/jds/jds.pdf}{pdf})

\end{itemize}
The description in terms of [[groupoid object in an (∞,1)-category]] is due to

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
see last remark of section 6.1.2 .

Wikipedia's definition (at time of this writing, and phrased in the language of [[homotopy theory]]) is rather a [[unitoid]] object in the $(\infty,1)$-category [[Top]].

\begin{itemize}%
\item MathOverflow: \href{http://mathoverflow.net/questions/16711/homotopy-associative-h-space-and-coh-space}{homotopy-associative-h-space-and-coh-space}

\end{itemize}
See also

\begin{itemize}%
\item [[John Adams]], \emph{Finite $H$-Spaces and Lie groups}, Journal of Pure and Applied Algebra 19 (1980) 1-8 (\href{http://www.maths.ed.ac.uk/~aar/papers/adamse8.pdf}{pdf})


\item [[John Adams]], \emph{The sphere, considered as an $H$-space mod $p$}, Quart. J. Math. Oxford. Ser. (2) vol 12 52-60 (\href{http://qjmath.oxfordjournals.org/content/12/1/52.citation}{pdf})



\end{itemize}
[[!redirects H-spaces]]

[[!redirects H-monoid]] [[!redirects H-monoids]]

[[!redirects H-group]] [[!redirects H-groups]]

[[!redirects H-ring]] [[!redirects H-rings]]

[[!redirects H-category]] [[!redirects H-categories]]

[[!redirects H-unitoid]] [[!redirects H-cogroup]] [[!redirects H-comonoid]]



\end{document}