\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*{Nonabelian Algebraic Topology}

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

This entry is about the book

\begin{itemize}%
\item [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], \emph{Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids} Tracts in Mathematics 15, European Mathematical Society , \href{http://groupoids.org.uk/nonab-a-t.html}{web}, from which the full pdf is available.

\end{itemize}
The publication details of the book are as follows:

ISBN 978-3-03719-083-8, DOI 10.4171/083

August 2011, 703 pages, hardcover, 17 x 24 cm.

98.00 \euro{}

(distribution via the AMS in October, 2011)

The EMS allows a full pdf (with hyperref) on Ronnie Brown's web page, \href{http://groupoids.org.uk/nonab-a-t.html}{web}.

This treats [[algebraic topology]] using tools of [[strict ∞-groupoid]]-theory: notably the traditional [[homological algebra]] use of [[chain complex]]es of abelian groups is generalized to [[crossed complex]]es, and emphasis is put on the notion of [[fundamental groupoid]] and its strict [[higher category theory|higher categorical]] generalizations to the cubical [[fundamental omega-groupoid]] of a filtered space over the bare [[homotopy group]]s of a space.

One of the main motivations for the development of Nonabelian Algebraic Topology was the observation that the [[Seifert-van Kampen theorem]] is most naturally understood as being not about [[homotopy group]]s, but about the cubical [[fundamental omega-groupoid]] of a filtered space and may be generalized to a [[higher homotopy van Kampen theorem]] this way.

The restriction to [[strict ∞-groupoid]]s/[[crossed complex]]es is still a severe restriction as compared to the full [[homotopy theory]] of [[topological space]]s but already more general than the strict and strictly abelian $\infty$-groupoids used in traditional [[algebraic topology]] in the guise of [[chain complex]]es of abelian groups. In terms of the [[cosmic cube]] of [[higher category theory]] the approach of \emph{Nonabelian algebraic topology} used here is somewhere half way between homology and homotopy theory; it in this border area that traditional accounts seem to most lacking, and are unable to cope well with the nonabelian second relative homotopy group of a pair of spaces. The start of the new approach is to replace this by a homotopy double groupoid of a pair of spaces, which allows an \emph{algebraic inverse to subdivision}. The philosophy behind the work is also given in the paper Modelling and Computing Homotopy Types: I

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

\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{contents_2}{Contents}\dotfill \pageref*{contents_2} \linebreak
\noindent\hyperlink{i_1_and_2dimensional_results}{I 1 and 2-dimensional results}\dotfill \pageref*{i_1_and_2dimensional_results} \linebreak
\noindent\hyperlink{crossed_complexes}{Crossed complexes}\dotfill \pageref*{crossed_complexes} \linebreak
\noindent\hyperlink{7_the_basics_of_crossed_complexes}{7 The basics of crossed complexes}\dotfill \pageref*{7_the_basics_of_crossed_complexes} \linebreak
\noindent\hyperlink{71_our_basic_categories_and_functors}{7.1 Our basic categories and functors}\dotfill \pageref*{71_our_basic_categories_and_functors} \linebreak
\noindent\hyperlink{71i_the_category_of_filtered_topological_spaces}{7.1.i The category of filtered topological spaces}\dotfill \pageref*{71i_the_category_of_filtered_topological_spaces} \linebreak
\noindent\hyperlink{71ii_modules_over_groupoids}{7.1.ii Modules over groupoids}\dotfill \pageref*{71ii_modules_over_groupoids} \linebreak
\noindent\hyperlink{71iii_the_category_of_crossed_complexes}{7.1.iii The category of crossed complexes}\dotfill \pageref*{71iii_the_category_of_crossed_complexes} \linebreak
\noindent\hyperlink{71iv_homotopy_and_homology_groups_of_crossed_complexes}{7.1.iv Homotopy and homology groups of crossed complexes}\dotfill \pageref*{71iv_homotopy_and_homology_groups_of_crossed_complexes} \linebreak
\noindent\hyperlink{FundamentalCrossedComplex}{7.1.v The fundamental crossed complex functor}\dotfill \pageref*{FundamentalCrossedComplex} \linebreak
\noindent\hyperlink{CrsdAndChainCplx}{7.4 Crossed complexes and chain complexes}\dotfill \pageref*{CrsdAndChainCplx} \linebreak
\noindent\hyperlink{AdjointModule}{7.4.i Adjoint module and augmentation module}\dotfill \pageref*{AdjointModule} \linebreak
\noindent\hyperlink{ChainOfCrossed}{7.4.iii The derived chain complex of a crossed complex}\dotfill \pageref*{ChainOfCrossed} \linebreak
\noindent\hyperlink{CrsFromCh}{7.4.v The right adjoint of the derived functor}\dotfill \pageref*{CrsFromCh} \linebreak
\noindent\hyperlink{8_the_higher_homotopy_van_kampen_theorem_and_its_applications}{8 The Higher Homotopy van Kampen Theorem and its applications}\dotfill \pageref*{8_the_higher_homotopy_van_kampen_theorem_and_its_applications} \linebreak
\noindent\hyperlink{ChainCplxOfFilteredSpace}{8.4 The chain complex of a filtered space and of a CW-complex}\dotfill \pageref*{ChainCplxOfFilteredSpace} \linebreak
\noindent\hyperlink{example_chains_on_the_simplex}{Example: Chains on the $n$-simplex}\dotfill \pageref*{example_chains_on_the_simplex} \linebreak
\noindent\hyperlink{9_tensor_products_and_homotopies_of_crossed_complexes}{9 Tensor products and homotopies of crossed complexes}\dotfill \pageref*{9_tensor_products_and_homotopies_of_crossed_complexes} \linebreak
\noindent\hyperlink{HomotopyAdditionSimplex}{9.9 The homotopy addition lemma for a simplex}\dotfill \pageref*{HomotopyAdditionSimplex} \linebreak
\noindent\hyperlink{SimpSetAndCrs}{9.10 Simplicial sets and crossed complexes}\dotfill \pageref*{SimpSetAndCrs} \linebreak
\noindent\hyperlink{DoldKanMap}{Remark 9.10.6 (Dold-Kan map and $\omega$-nerve)}\dotfill \pageref*{DoldKanMap} \linebreak
\noindent\hyperlink{10_resolutions}{10 Resolutions}\dotfill \pageref*{10_resolutions} \linebreak
\noindent\hyperlink{11_the_cubical_classifying_space_of_a_crossed_complex}{11 The cubical classifying space of a crossed complex}\dotfill \pageref*{11_the_cubical_classifying_space_of_a_crossed_complex} \linebreak
\noindent\hyperlink{12_nonabelian_cohomology_spaces_groupoids}{12 Nonabelian cohomology: spaces, groupoids}\dotfill \pageref*{12_nonabelian_cohomology_spaces_groupoids} \linebreak
\noindent\hyperlink{cubical_groupoids}{Cubical $\omega$-groupoids}\dotfill \pageref*{cubical_groupoids} \linebreak
\noindent\hyperlink{13_the_algebra_of_crossed_complexes_and_cubical_groupoids}{13 The algebra of crossed complexes and cubical $\omega$-groupoids}\dotfill \pageref*{13_the_algebra_of_crossed_complexes_and_cubical_groupoids} \linebreak
\noindent\hyperlink{14_cubical_homotopy_groupoid}{14 Cubical homotopy groupoid}\dotfill \pageref*{14_cubical_homotopy_groupoid} \linebreak
\noindent\hyperlink{CubicalDoldKan}{14.8 The cubical Dold-Kan theorem}\dotfill \pageref*{CubicalDoldKan} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{history}{}\subsection*{{History}}\label{history}

Some comments from [[Ronnie Brown]] himself:

\begin{quote}%
I hope it is helpful to relate my experiences from the 1960s and later with [[nonabelian cohomology]].

In writing my book on [[topology]] in the 1960s, I got offended by having to make a detour to get the [[fundamental group]] of the circle, and then was attracted by [[Paul Olum]]`s paper referenced below. I extended Olum's work to a [[Mayer-Vietoris sequence|Mayer?Vietoris type sequence]] in the second paper below, and this enabled one to compute the fundamental group of, for example, a [[wedge product|wedge]] of circles.

(I use an MV sequence in \emph{Topology and Groupoids} in connection with pullbacks of [[covering space]]s.)

So I decided to use this account for the book, thus giving students the advantage, it seemed, of an introduction to [[cohomology|cohomological]] ideas.

The problem was that the account when written in detail came to 30 pages (or maybe 40) and when looked at in the cold light of day seemed incredibly boring (a full account is different from Olum's research account).

I was at the time looking for exercises and came across [[Philip Higgins]]` paper on [[presentation]]s of groupoids, which used free products with amalgamation of groupoids. So I decided to give an exercise on the fundamental groupoid of a union. Then I felt I ought to write out a solution. When I had done this, it seemed streets ahead in exposition of all that [[nonabelian cohomology]] stuff and moreover, when souped up to the \emph{[[fundamental groupoid]] on a set of base points} , gave results not reachable by the MV sequence; for example you could not with the MV sequence deduce the \emph{precise calculation} of the fundamental group of a union of two open sets whose intersection had say 150 path components. (This anomaly is also significant, in illustrating the limitations of exact sequences.)

So I decided to switch to an exposition of groupoids in 1-dimensional [[homotopy theory]] (also spurred by a meeting with [[George Mackey]] in 1967 where he told me of his work on ergodic groupoids, which is now seen as a preliminary to [[noncommutative geometry|Noncommutative Geometry]]).

It occurred to me that if one could come to the groupoid idea from two distinct directions, then there was likely to be more in this than met the eye. At the same time, an examination of the proof of the [[van Kampen theorem]] for groupoids, suggested that the theorem should have an extension to all dimensions, if one could define homotopy gadgets with the right properties. Another stimulus was the proof (used in the book) by [[Frank Adams]] (circulated in handwritten lecture notes) of the [[cellular approximation theorem]], which had analogies to parts of the van Kampen proof, but failed to get algebraic results because, apparently, of the lack of an appropriate algebraic gadget in dimension $n \gt 1$.

It took 9 years to find such a gadget in dimension $2$, and another 3 to get them in all dimensions, in work with [[Philip Higgins]].

It seemed to me unfortunate that this work aroused the opposition, for reasons never explained to me, of Frank Adams, who told people the whole programme was ``ridiculous''. His opinion became the opposite only when I told him (1985?) of the extension to the non simply connected case of the Blakers-Massey description of $\pi_3$ of a triad, using the nonabelian tensor product (work with [[Jean-Louis Loday]]).

The higher order van Kampen theorems, and the often nonabelian calculations which result, have not been obtained by cohomological methods, but only by working directly with structures appropriate to the geometry of higher homotopies, i.e. forms of strict [[n-fold category|multiple groupoids]]. This confirms the comment of [[Philip Hall]], Philip Higgins' supervisor, that one should not try to force the geometry into a given algebraic mode, but search for the algebra which models the geometry. So it seems to me that algebraic topology has been mainly restricted to, or not got out of, the single base point and ``group'', not ``groupoid'', mode, nor appreciated the possibilities of [[colimit]] type theorems in algebraic (and geometric?) topology -- no algebraic or geometric topology text (except mine!) mentions the higher order van Kampen work with Philip Higgins.

You can also see this restriction in the contrast between the unsymmetrical, choice laden, definition of the second relative homotopy group, with its compositions in one direction (recall the limitations of ``Lineland'' described in ``Flatland'') and the definition of the fundamental [[double groupoid]] of a pointed pair of spaces $\rho_2(X,A)$, with its compositions in $2$ directions. This contrast gets more significant in higher dimensions.

For all these reasons, my inclination is to look for the applications of the ``appropriate'' (whatever that is!) structures rather than cohomology with coefficients in such structures, where lots of detail is likely to get lost. Also, in making calculations it is convenient to work with strict algebraic structures, where the notion of colimit is more comprehensible. Even there, it has been a problem to make say colimit calculations with [[crossed module]]s into a symbolic computer algebra format. See the work by [[Chris Wensley]] listed below.

These results could not have been obtained without the intuitions on multiple compositions easily allowed by a cubical approach.

One of the key observations for this programme was that one could define a strict homotopy double groupoid for a \emph{pointed pair of spaces}, and that this was closely related to the well known fundamental crossed module of a pair of spaces, first considered by J.H.C. [[Whitehead]]. His paper listed below was a key source of ideas.

The natural extension of this observation is to construct a [[strict cubical omega-groupoid|strict cubical ∞-groupoid]] $\rho X_*$ of a \emph{[[filtered space]]} $X_*$, and find its relation to the quite classical homotopically defined fundamental crossed complex functor $\Pi: (filtered spaces) \to (crossed complexes)$. The proofs here are non trivial. By proving using $\rho$ a colimit theorem for $\Pi$ one can shortcut [[singular homology]], and obtain old and new results in algebraic topology, including some explicit calculations of homotopy groups, even as modules over the fundamental group. This working with filtered space is not unreasonable since they abound. For example, classifying spaces often come with convenient filtrations, as do [[geometric realisation]]s of [[simplicial set|simplicial]] or [[cubical set|cubical]] sets. These ideas generalise of course to [[multifiltered space]]s or $n$-cubes of spaces. It is not so clear that one \emph{must} work with a kind of bare topological space, and so have little handle on which to construct invariants, except say by first taking a singular complex, or using multipaths.

The main idea of the [[higher homotopy van Kampen Theorem]]s is to model algebraically the gluing of homotopy types, or limited models of such.

An indication of a beginnings of a ech type approach to nonabelian cohomology using groupoids and crossed complexes is given in the new book, Chapter 12. This has not been developed in terms of [[sheaf theory]].

Another big gap in comparison with traditional algebraic topology is [[intersection theory]] and [[Poincare duality]], although the (quite complicated) machinery of tensor products is available in the crossed complex context.

An obvious gap is also that of extending [[Grothendieck]]`s work on the fundamental group!


\end{quote}
\hypertarget{contents_2}{}\subsection*{{Contents}}\label{contents_2}

\hypertarget{i_1_and_2dimensional_results}{}\subsubsection*{{I 1 and 2-dimensional results}}\label{i_1_and_2dimensional_results}

\begin{itemize}%
\item [[groupoid]]


\item [[2-groupoid]]


\item [[2-group]]


\item [[crossed module]]


\item [[fundamental groupoid]]



\end{itemize}
\hypertarget{crossed_complexes}{}\subsubsection*{{Crossed complexes}}\label{crossed_complexes}

\hypertarget{7_the_basics_of_crossed_complexes}{}\paragraph*{{7 The basics of crossed complexes}}\label{7_the_basics_of_crossed_complexes}

\begin{itemize}%
\item [[crossed complex]]

\end{itemize}
\hypertarget{71_our_basic_categories_and_functors}{}\paragraph*{{7.1 Our basic categories and functors}}\label{71_our_basic_categories_and_functors}

\hypertarget{71i_the_category_of_filtered_topological_spaces}{}\paragraph*{{7.1.i The category of filtered topological spaces}}\label{71i_the_category_of_filtered_topological_spaces}

\hypertarget{71ii_modules_over_groupoids}{}\paragraph*{{7.1.ii Modules over groupoids}}\label{71ii_modules_over_groupoids}

A \emph{[[module over a groupoid]]} is a collection of [[abelian group]]s equipped with a linear [[action]] by a [[groupoid]].

\begin{udef}
\textbf{(module over a groupoid)}

Let $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ be a [[groupoid]]. A \textbf{module over the groupoid} $\mathcal{G}$ is a collection $\{N_x\}_{x \in \mathcal{G}_0}$ of [[abelian group]]s equipped with a collection of maps

\begin{displaymath}
N_x \times \mathcal{G}(x,y) \to N_y
\end{displaymath}
that are linear and respect the groupoid composition in the obvious way.

\end{udef}
\hypertarget{71iii_the_category_of_crossed_complexes}{}\paragraph*{{7.1.iii The category of crossed complexes}}\label{71iii_the_category_of_crossed_complexes}

\begin{itemize}%
\item [[crossed complex]]

\end{itemize}
\hypertarget{71iv_homotopy_and_homology_groups_of_crossed_complexes}{}\paragraph*{{7.1.iv Homotopy and homology groups of crossed complexes}}\label{71iv_homotopy_and_homology_groups_of_crossed_complexes}

(\ldots{})

\hypertarget{FundamentalCrossedComplex}{}\paragraph*{{7.1.v The fundamental crossed complex functor}}\label{FundamentalCrossedComplex}

The notion of [[fundamental groupoid]] of a [[topological space]] generalizes to a notion of [[fundamental ∞-groupoid]]. There is a strict version of this (which loses some information): the fundamental strict $\infty$-groupoid. In a [[filtered space]] $X_*$ one can consider the variant where the [[k-morphism]]s of the fundamental $\infty$-groupoid are constrained to lie in $X_k$. The \emph{fundamental crossed complex} of a filtered space is the equivalent [[crossed complex]] incarnation of the fundamental strict $\infty$-groupoid of a filtered space.

\begin{udef}
\textbf{(fundamental crossed complex)}

Let $X_\bullet$ be a [[filtered space]].

Write $\Pi_1(X_1,X_0)$ for the subgroupoid of the [[fundamental groupoid]] $\Pi_1(X_1)$ of $X_1$ on objects that are in $X_0$.

The \textbf{fundamental crossed complex} $\Pi X_*$ of $X$ is the [[crossed complex]]

with

\begin{displaymath}
(\Pi X_*)_1 = \Pi_1(X_1,X_0)
\end{displaymath}
and

\begin{displaymath}
(\Pi X_*)_n
  :=
  \coprod_{x \in X_0}
  \pi_n(X_n, X_{n-1}, x)
  \;\;\;\;
  for n \geq 2
  \,,
\end{displaymath}
where $\pi_n(X_n, X_{n-1}, x)$ is the [[relative homotopy group]] obtained by equivalence classes of maps from the pointed $n$-disk into $X$ such that the disk lands in $X_n$, its boundary in $X_{n-1}$ and its basepoint on $x$.

\end{udef}
See also section 1 of

\begin{itemize}%
\item [[Ronnie Brown]], [[Rafael Sivera]], \emph{Normalization of the fundamental crossed complex of a simplicial set} (\href{http://arxiv.org/abs/math/0611728}{arXiv:math/0611728}, \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.63.9955&rep=rep1&type=pdf}{pdf})

\end{itemize}
\begin{uexample}
\textbf{(fundamental crossed complex of the $n$-simplex)}

The topological $n$-[[simplex]] $\Delta^n$ is canonically a [[filtered space]] with $(\Delta^n)_k$ being the union of its $k$-faces.

Then we have that $\Pi_1((\Delta^n)_1, (\Delta^n)_0)$ is the groupoid whose objects are the $n+1$ vertices of $\Delta^n$ and which has precisely one morphism $x_i \to x_j$ for each ordered pair $x_i,x_j \in (\Delta^n)_0$ (all of them being [[isomorphism]]s)

\begin{displaymath}
\Pi_1((\Delta^2)_1,(\Delta^2)_0) = 
  \left\{
     \itexarray{
     && x_1
     \\
     & \nearrow\swarrow && \searrow \nwarrow
     \\
    x_0 &&\stackrel{\leftarrow}{\to}&& x_2
    }
  \right\}
  \,.
\end{displaymath}
At any $x_i$ the relative homotopy group $\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i)$ is a group on the set of 2-faces that have $x_i$ as a 0-face: there is a unique homotopy class of disks in $\Delta^n$ that sits in the 2-faces $(\Delta^n)_2$, whose base point is at $x_j$ and whose boundary runs along the boundary of a given 2-face of $\Delta^n$.

So (using the equivalence of crossed complexes with strict $\omega$-groupoids) for instance $\Pi \Delta^2$ is generated from $\Pi_1((\Delta^2)_1,(\Delta^2)_0)$ as above and a 2-cell

\begin{displaymath}
\itexarray{
     && x_1
     \\
     & \swarrow &\Downarrow& \nwarrow
     \\
    x_0 &&\to&& x_2
    }
\end{displaymath}
under [[whiskering]] and composition. For instance [[whiskering]] this with $x_1 \to x_2$ yields the 2-morphism

\begin{displaymath}
\itexarray{
     && x_1
     \\
     & \swarrow &\swArrow& \searrow
     \\
    x_0 &&\to&& x_2
    }
  \,.
\end{displaymath}
One sees that $\Pi \Delta^2$ is the strict groupoidification of the second [[oriental]].

Generally, $\Pi \Delta^n$ is the $n$-groupoid freely generated from $k$-morphisms for each $k$-face of $\Delta^n$.

\end{uexample}
\hypertarget{CrsdAndChainCplx}{}\paragraph*{{7.4 Crossed complexes and chain complexes}}\label{CrsdAndChainCplx}

\begin{udef}
\textbf{(groupoid module chain complexes)}

Write $Chn$ for the [[category]] of [[chain complex]]es of modules over a [[groupoid]].

\end{udef}
This is Def. 7.4.1.

\begin{udef}
\textbf{(groupoid module chain complexes)}

Write $Crs$ for the [[category]] of [[crossed complex]]es.

\end{udef}
\hypertarget{AdjointModule}{}\paragraph*{{7.4.i Adjoint module and augmentation module}}\label{AdjointModule}

\begin{udef}
Given a [[module over a groupoid]] $(N,\mathcal{G})$, the \textbf{semidirect product groupoid} $\mathcal{G} \ltimes N$ has the same objects as $\mathcal{G}$ and morphisms

\begin{displaymath}
(\mathcal{G} \ltimes N)(p,q) = \mathcal{G}(p,q) \ltimes N(q)
\end{displaymath}
with composition given by the action of $\mathcal{G}$ on $N$.

\end{udef}
This is def. 7.4.5

\begin{udef}
\textbf{(covering morphism)}

For $(t : N \to \mathcal{G}_0,\mathcal{G})$ a [[module over a groupoid]], write $P(N,\mathcal{G})$ for the groupoid $\mathcal{G}$ pulled back to the underlying set of $N$:

an object of $P(N,\mathcal{G})$ is an element in $N$ and a morphism $n_1 \to n_2$ is a morphism $\mathcal{G}(t(n_1),t(n_2))$.

\end{udef}
This is def. 7.4.9.

\hypertarget{ChainOfCrossed}{}\paragraph*{{7.4.iii The derived chain complex of a crossed complex}}\label{ChainOfCrossed}

\begin{udef}
\textbf{(chain complex from a crossed complex)}

Define a functor $\nabla : Crs \to Chn$ from [[crossed complex]]es to [[modules over groupoids]] as follows:

For $C$ a [[crossed complex]] we set for $n \geq 3$

\begin{displaymath}
(\nabla C)_n := C_n \;\;\;\; for n \geq 3
\end{displaymath}
and for $n \leq 2$ it is given by \ldots{}

\end{udef}
This is definition 7.4.20.

\hypertarget{CrsFromCh}{}\paragraph*{{7.4.v The right adjoint of the derived functor}}\label{CrsFromCh}

We describe a construction of a [[crossed complex]] from a [[chain complex]] of [[modules over a groupoid]] $(A_n, \mathcal{H})$. As a special case it in particular gives an map of ordinary [[chain complex]]es of [[abelian group]]s into the category of [[crossed complex]]es, and hence into [[strict ∞-groupoid]]s.

Recall the definition of the \hyperlink{AdjointModule}{semidirect product groupoid} $\mathcal{H} \ltimes A_n$.

\begin{udef}
\textbf{(crossed complex from a chain complex)}

For $A$ a [[chain complex]] of [[modules over a groupoid]] $\mathcal{H}$, let $\Theta A \in Crs$ be the [[crossed complex]]

\begin{displaymath}
\Theta A := \kappa^* \Theta' A
  \,,
\end{displaymath}
where

\begin{displaymath}
\Theta' A :=
  \left[
    A_n 
      \stackrel{\partial_n}{\to}
    A_{n-1}
     \stackrel{}{\to}
     \cdots
     \stackrel{}{\to}
    A_{3}
    \stackrel{\partial_3}{\to}
    A_2
    \stackrel{(0,\partial_2)}{\to}
    \mathcal{H}\ltimes A_1
  \right]
\end{displaymath}
and where

\begin{displaymath}
\kappa : P(A_0, \mathcal{H}) \to \mathcal{H} \ltimes A_0
\end{displaymath}
is the \hyperlink{AdjointModule}{canonical covering morphism} from above.

\begin{displaymath}
\itexarray{
    \cdots \to & 
    (\Theta A)_3
    &\to&
    (\Theta A)_2
    &\to&
    (\Theta A)_1
    &\to&
    P(A_0, \mathcal{H})
    \\
    & \downarrow && \downarrow && \downarrow && \downarrow
    \\
    \cdots \to & 
    A_3
    &\stackrel{\partial_3}{\to}&
    A_2
    &\stackrel{(0,\partial_2)}{\to}&
    \mathcal{H} \ltimes A_1
    &\stackrel{(1, \partial_1)}{\to}&
    \mathcal{H} \ltimes A_0    
  }
  \,.
\end{displaymath}
Here $\mathcal{H} \ltimes A_1$ acts on $A_n$ for $n \geq 2$ via the projection $\mathcal{H} \ltimes A_1 \to \mathcal{H}$, i.e. $A_1$ acts trivially. (\ldots{})

Finally set $\Theta(A)_0 := A_0$.

\end{udef}
We spell out what this boils down to explicitly.

\textbf{Explicit description}

Let $A_\bullet$ be a [[chain complex]] of [[modules over a groupoid|modules over the groupoid]] $\mathcal{H}$. Then the [[crossed complex]] $\Theta(A)$ is the following.

\begin{itemize}%
\item Its set of objects is $\Theta(A)_0 = A_0$.

Remember that $A_0$ itself is a module over $\mathcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0)$, so that $A_0 = \coprod_{p \in \mathcal{H}_0} (A_0)_p$.


\item For $x \in (A_0)_p$ and $y \in (A_0)_q$ a morphism in $\Theta(A)_1$ from $x$ to $y$ is labeled by $h \in \mathcal{H}_1$ and $a \in (A_1)_q$

\begin{displaymath}
x \stackrel{(h,a)}{\to} (y = \rho(h)(x) - \partial a)
  \,,
\end{displaymath}
where $\rho$ denotes the [[action]] of $\mathcal{H}$ on $A_0$.

The composition law is given by

\begin{displaymath}
\itexarray{
     && y
     \\
     & {}^{\mathllap{(h_1, a_1)}}\nearrow && \searrow^{\mathrlap{(h_2,a_2)}}
     \\
     x &&\stackrel{(h_1 \circ h_2, \rho(h_2)(a_1) + a_2)}{\to}&& z
  }
  \,.
\end{displaymath}

\item For $k \geq 2$ the family of groups $\Theta(A)_k$ is over $x \in (A_0)_p$ the group $(A_k)_q$

\begin{displaymath}
\Theta(A)_{k \geq 2} = \coprod_{p \in \mathcal{H}_0} \coprod_{x\in (A_0)_q} (A_k)q
\end{displaymath}

\item The boundary maps and actions are the obvious ones\ldots{}



\end{itemize}
\begin{uexample}
\textbf{(ordinary abelian chain complex as crossed complex)}

Let $C_\bullet$ be an ordinary [[chain complex]] of abelian groups, i.e. a chain complex of [[module over a groupoid|modules over the trivial groupoid]].

Then $(\Theta C)_1$ is the groupoid with objects $C_0$ and morphisms $\{x \stackrel{b}{\to} (x + \partial b)\}$. And for $n \geq 2$ we have that $(\Theta C)_n$ is $\coprod_{x \in C_0} C_n$.

\end{uexample}
\begin{uprop}
These form a pair of [[adjoint functor]]s

\begin{displaymath}
(\nabla \dashv \Theta)
  : 
  Chn
  \stackrel{\overset{\nabla}{\leftarrow}}{\underset{\Theta}{\to}}
  Crs
\end{displaymath}
where\ldots{}

\end{uprop}
This is proposition 7.4.29.

(\ldots{})

\hypertarget{8_the_higher_homotopy_van_kampen_theorem_and_its_applications}{}\paragraph*{{8 The Higher Homotopy van Kampen Theorem and its applications}}\label{8_the_higher_homotopy_van_kampen_theorem_and_its_applications}

\begin{itemize}%
\item [[higher homotopy van Kampen theorem]]

\end{itemize}
\hypertarget{ChainCplxOfFilteredSpace}{}\paragraph*{{8.4 The chain complex of a filtered space and of a CW-complex}}\label{ChainCplxOfFilteredSpace}

Let all [[topological space]]s $X$ in the following by [[Hausdorff space]]s that admit a [[universal cover]]. $\hat X$.

\begin{udef}
\textbf{(homology chain complex of a filtered space)}

For $X = (X_\bullet)$ a [[filtered space]] define a [[chain complex]] of [[modules over a groupoid]] $\mathcal{C}_\bullet(X)$ as follows.

The [[groupoid]] $\mathcal{G} := \Pi_1(X,X_0)$ is the full subgroupoid of the [[fundamental groupoid]] of $X$ on points in $X_0$.

For $x_0 \in X$ let $\hat X(x_0) := \coprod_y \Pi_1(y,x_0)$ be the standard model for the [[universal cover]] of $X$ in terms of homotopy classes of paths into $x_0$.

For all $x \in X_0 = Obj(\Pi_1(X,X_0))$ take the [[module over a groupoid|modules over]] $\Pi_1(X,X_0)$ to be the [[relative homology group]]s

\begin{displaymath}
(\mathcal{C}_0 X)_x := H_0(\hat X_0(x))
\end{displaymath}
and for $n \geq 1$

\begin{displaymath}
(\mathcal{C}_n X)_x := H_n(\hat X_n(x),\hat X_{n-1}(x) )
  \,.
\end{displaymath}
The action of $\Pi_1(X,X_0)$ on this is the evident one induced by composition of paths.

This extends to a [[functor]]

\begin{displaymath}
\mathcal{C}_\bullet : FTop \to Chn
  \,.
\end{displaymath}
\end{udef}
This is \emph{def 8.4.1}

The next proposition asserts that this notion of chain complex of a filtered topological space is reproduced by the combination of

\begin{itemize}%
\item the \hyperlink{FundamentalCrossedComplex}{fundamental crossed complex} $\Pi X_\bullet$


\item and the \hyperlink{ChainOfCrossed}{chain complex of a crossed complex} $\nabla \Pi X_\bullet$.



\end{itemize}
\begin{uprop}
If the [[filtered space]] $X_\bullet$ is [[connected space|connected]] then there is a [[natural isomorphism]]

\begin{displaymath}
\mathcal{C}_\bullet X \simeq \nabla \Pi X
  \,.
\end{displaymath}
\end{uprop}
This is \emph{proposition 8.4.2} . Use the [[Hurewicz theorem|relative Hurewicz theorem]] to translate from [[homotopy group]]s to [[homology group]]s.

\hypertarget{example_chains_on_the_simplex}{}\paragraph*{{Example: Chains on the $n$-simplex}}\label{example_chains_on_the_simplex}

\begin{uexample}
\textbf{(chains on the $n$-simplex)}

Consider $X = \Delta^n$, the standard topological $n$-[[simplex]] regarded as a [[filtered space]] with the union of its $k$-faces in degree $k$.

Notice that since $\Delta^n$ is a [[simply connected space]] in this case we have that for each basepoint $x \in (\Delta^n)_0$ the universal cover $\hat X_{x} = X$ coincices with $X$.

We have that

\begin{displaymath}
\mathcal{C}_\bullet \Delta^n \simeq \nabla \Pi \Delta^n
  \simeq
   N_\bullet \Delta[n]
\end{displaymath}
is, over each vertex $x \in (\Delta^n)_0$, the [[Dold-Kan correspondence|normalized chain complex]] of [[cochain on a simplicial set|chains on the simplicial set]] $\Delta[n]$

\begin{displaymath}
\mathcal{C}_0 \Delta^n = \mathbb{Z}^{n+1}
  \;\;\;\;\;
  (\mathcal{C}_0 \Delta^n)_x = \mathbb{Z}
\end{displaymath}
\begin{displaymath}
(\mathcal{C}_1 \Delta^n)_{x} = \mathbb{Z}^n
\end{displaymath}
etc.

\begin{displaymath}
(\mathcal{C}_n \Delta^n)_x = \mathbb{Z}
  \,.
\end{displaymath}
We have moreover that $\Pi_1(\Delta^n, (\Delta^n)_0)$ is the [[codiscrete groupoid]] on $n+1$ objects. It acts on the $\mathcal{C}_k(\Delta^n)$ by identity maps

\begin{displaymath}
(x_i \to x_j) : (\mathcal{C}_{k} \Delta^n)_{x_i} 
  \stackrel{=}{\to}
  (\mathcal{C}_{k} \Delta^n)_{x_j}
  \,.
\end{displaymath}
It follows in particular that for $D_\bullet$ an ordinary [[chain complex]] of [[abelian group]]s regarded as a complex of [[modules over a groupoid]] in the trivial way, morphisms of modules over groupoids

\begin{displaymath}
\mathcal{C}_\bullet \Delta^n \to D
\end{displaymath}
are canonically identified with morphisms of ordinary chain complexes of abelian groups

\begin{displaymath}
N_\bullet \Delta[n] \to D
  \,.
\end{displaymath}
For more on this see \hyperlink{DoldKanMap}{Dold-Kan map and omega-nerve}.

\end{uexample}
\hypertarget{9_tensor_products_and_homotopies_of_crossed_complexes}{}\paragraph*{{9 Tensor products and homotopies of crossed complexes}}\label{9_tensor_products_and_homotopies_of_crossed_complexes}

\hypertarget{HomotopyAdditionSimplex}{}\paragraph*{{9.9 The homotopy addition lemma for a simplex}}\label{HomotopyAdditionSimplex}

For $\Delta^n$ the topological $n$-simplex regarded as a [[filtered space]] in the canonical way, the fundamental crossed complex $\Pi X^n$ is a groupoid-version of the $n$-[[oriental]]: the free [[strict ∞-groupoid]] on a single $n$-simplex.

\hypertarget{SimpSetAndCrs}{}\paragraph*{{9.10 Simplicial sets and crossed complexes}}\label{SimpSetAndCrs}

By the discussion at \hyperlink{HomotopyAdditionSimplex}{The homotopy addition lemma for a simplex} the \hyperlink{FundamentalCrossedComplex}{fundamental crossed complex} $\Pi \Delta^n$ plays the role of the free strict $n$-groupoid on the $n$-[[simplex]].

The [[cosimplicial object|cosimplicial]] $\infty$-groupoid

\begin{displaymath}
\Pi \Delta^\bullet : 
  \Delta \to Crs \simeq Str \infty Grpd
\end{displaymath}
induced by the discussion at [[nerve and realization]] a simplicial [[nerve]] operation on [[strict ∞-groupoid]] -- an [[∞-nerve]]:

\begin{udef}
\textbf{(simplicial nerve)}

Let $C$ be a [[crossed complex]]. Its \textbf{simplicial [[nerve]]} $N^\Delta C \in$ [[sSet]] is

\begin{displaymath}
(N^\Delta C)_n := Crs(\Pi \Delta^n, C)
\end{displaymath}
\end{udef}
This is definition 9.10.2.

\hypertarget{DoldKanMap}{}\paragraph*{{Remark 9.10.6 (Dold-Kan map and $\omega$-nerve)}}\label{DoldKanMap}

\begin{uprop}
\textbf{(Dold-Kan map)}

For $D \in Chn$ a [[chain complex]] (of [[abelian groups]]) regarded as a chain complex of [[modules over a groupoid|modules over]] the trivial groupoid, we may regard it as a [[crossed complex]] $\Theta D$ as described at \hyperlink{CrsFromCh}{Crossed complex from chain complex}, hence as a [[strict ∞-groupoid]].

The [[∞-nerve]] $N^\Delta \Theta D \in$ [[sSet]] (described in \hyperlink{SimpSetAndCrs}{Crossed complexes and simplicial sets}) of this is the [[Kan complex]] underlying the image of $D$ under the [[Dold-Kan correspondence]] $Chn \to sAb$.

\end{uprop}
\begin{proof}
By definition we have

\begin{displaymath}
N^\Delta (\Theta D)  := Crs(\Pi \Delta^\bullet, \Theta D)
  \,.
\end{displaymath}
By [[adjunction]] $(\Pi \dashv \Theta)$ with the \hyperlink{CrsFromCh}{Theta-map} this is equivalently

\begin{displaymath}
\cdots \simeq Chn( \nabla \Pi \Delta^\bullet, D)
\end{displaymath}
Using the propositions and examples discussed at \hyperlink{ChainCplxOfFilteredSpace}{Chain complex of a filtered space} we have that $\nabla \Pi \Delta^n$ is standard normalized [[chain complex]] $N_\bullet \Delta[n]$ of chains on the simplicial $n$-[[simplex]] as discussed at [[cochain on a simplicial set|chains on a simplicial set]] and [[Dold-Kan correspondence]], but regarded as a complex of [[modules over a groupoid|modules over the groupoid]] $\Pi_1(\Delta^n, (\Delta^n)_0)$. But since the groupoid action on $D$ is trivial, the above is equivalent to

\begin{displaymath}
\cdots \simeq Chn( N_\bullet \Delta^\bullet , D) 
  \,.
\end{displaymath}
\end{proof}
This appears as \emph{remark 9.10.6} together with its \emph{footnote 116} .

\begin{uremark}
In the [[cosmic cube]] of [[higher category theory]] this realizes two edges

\begin{displaymath}
\itexarray{
    ChainCplx &\stackrel{\Theta}{\hookrightarrow}&
    CrossedCplx
    &\stackrel{N^\Delta}{\hookrightarrow}&
    KanCplx
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    StrAb Str\infty Grpd
    &\hookrightarrow&
    Str \infty Grpd
    &\hookrightarrow&
    \infty Grpd
  }
\end{displaymath}
including [[strict ∞-groupoid]]s with strict abelian [[∞-group]]-structure -- modeled as [[chain complex]]es of [[abelian group]]s -- into [[strict ∞-groupoid]]s -- modeled as [[crossed complex]]es -- into all [[∞-groupoid]]s -- modeled as [[Kan complex]]es. The composite is the map $Ch_\bullet \to sAb \to KanCplx$ to [[simplicial group]]s from the [[Dold-Kan correspondence]].

\end{uremark}
\hypertarget{10_resolutions}{}\paragraph*{{10 Resolutions}}\label{10_resolutions}

\hypertarget{11_the_cubical_classifying_space_of_a_crossed_complex}{}\paragraph*{{11 The cubical classifying space of a crossed complex}}\label{11_the_cubical_classifying_space_of_a_crossed_complex}

\hypertarget{12_nonabelian_cohomology_spaces_groupoids}{}\paragraph*{{12 Nonabelian cohomology: spaces, groupoids}}\label{12_nonabelian_cohomology_spaces_groupoids}

\begin{itemize}%
\item [[nonabelian cohomology]]

\end{itemize}
\hypertarget{cubical_groupoids}{}\subsubsection*{{Cubical $\omega$-groupoids}}\label{cubical_groupoids}

\hypertarget{13_the_algebra_of_crossed_complexes_and_cubical_groupoids}{}\paragraph*{{13 The algebra of crossed complexes and cubical $\omega$-groupoids}}\label{13_the_algebra_of_crossed_complexes_and_cubical_groupoids}

\hypertarget{14_cubical_homotopy_groupoid}{}\paragraph*{{14 Cubical homotopy groupoid}}\label{14_cubical_homotopy_groupoid}

\hypertarget{CubicalDoldKan}{}\paragraph*{{14.8 The cubical Dold-Kan theorem}}\label{CubicalDoldKan}

See also [[Dold-Kan correspondence]].

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

For an extensive list of relevant publications see

\begin{itemize}%
\item [[Ronnie Brown]], \emph{publication list} (\href{http://groupoids.org.uk/publicfull.html}{web})

\end{itemize}
Some selected references are:

\begin{enumerate}%
\item Olum, P., \emph{Non-abelian cohomology and van Kampen's theorem}, Ann. Math. 68 (1958) 658--667.


\item Brown, R., \emph{On a method of P. Olum}, J. London Math. Soc. 40 (1965) 303--304.


\item Brown, R., \emph{Elements of Modern Topology}, McGraw Hill, Maidenhead, 1968.


\item Brown, R., \emph{Topology and Groupoids}, Booksurge, 2006.


\item Higgins, P.J., \emph{Presentations of groupoids, with applications to groups}, Proc. Camb. Phil. Soc., 60 (1964) 7--20.


\item Brown, R. and Higgins, P.J., \emph{On the connection between the second relative homotopy groups of some related spaces}, Proc. London Math. Soc.(3) 36 (1978) 193--212.


\item Brown, R. and Higgins, P.J., \emph{Colimit theorems for relative homotopy groups}, J. Pure Appl. Algebra 22 (1981) 11--41.


\item Whitehead, J.H.C., \emph{Combinatorial Homotopy II}, Bull. Amer. Math. Soc., 55 (1949), 453--496.


\item Brown, R. \emph{Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems}, in Handbook of Algebra 6, Edited M. Hazewinkel, Elsevier, 2009.


\item Wensley, C.D. and Alp, M., XMOD, a GAP share package for computation with crossed modules, \emph{GAP Manual}, (1997), 1355--1420.


\item Brown, R., Higgins, P.J., and Sivera, R., \emph{Nonabelian Algebraic Topology: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids}, EMS Tracts in Mathematics, Vol. 15, (Autumn 2010).



\end{enumerate}
[[!redirects nonabelian algebraic topology]] [[!redirects Nonabelian algebraic topology]] [[!redirects Nonabelian Algebraic Topology]]

category: reference



\end{document}