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\begin{document}

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\section*{simplicial T-complex}

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

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

[[!include homotopy - contents]]

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

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{simplicial_complex_the_definition}{SImplicial $T$-complex: the definition}\dotfill \pageref*{simplicial_complex_the_definition} \linebreak
\noindent\hyperlink{results}{Results}\dotfill \pageref*{results} \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}

A \textbf{simplicial T-complex} is a [[Kan complex]] equipped with a \emph{choice} of [[horn]] fillers, which satisfy certain `equations'. It is thus a special case of an [[algebraic Kan complex]].

There is quite a difference between the [[Kan complex]] structure

\begin{enumerate}%
\item of the [[nerve|nerve]] $N(\mathcal{G})$ of a [[groupoid]], $\mathcal{G}$,


\item and that of, say, a [[path infinity-groupoid|singular complex]] $Sing X$ of some [[topological space]] $X$.



\end{enumerate}
In the first, if we are given a $(n,i)$-[[horn]], then there is \emph{exactly one} $n$-simplex in $Ner(G)$, since the $(n,i)$-horn has a chain of $n$-composable arrows of $G$ in it (at least unless $(n,i) = (2,0)$ or $(2,2)$, which cases are slightly different) and that chain gives the required $n$-simplex. In other words, there is a `canonical' filler for any horn. In $Sing(X)$, there will usually be many fillers; however the fact that this simplicial set is Kan is a property of retractions on standard simplices, and is not specifically a property of the space $X$ - that is the basic intuition.

Abstracting, in part, from this idea, Brown and Higgins developed the idea of a [[cubical T-complex]]. This was a [[cubical set|cubical set]] with in each dimension $n$, a subset of the $n$-[[cube]]s being declared \emph{`thin'}. The term was adopted to indicate that they, somehow, were of lower dimension than they looked to be. The theory was initiated in a simplicial context in the 1977 Bangor thesis of Keith Dakin listed below, and used by Brown and Higgins who showed that cubical $T$-complexes were equivalent to [[crossed complex|crossed complexes]]. The corresponding simplicial $T$-complex theory was further developed in the 1978 Bangor thesis of Nick Ashley, (see below for publication).

\hypertarget{simplicial_complex_the_definition}{}\subsection*{{SImplicial $T$-complex: the definition}}\label{simplicial_complex_the_definition}

\begin{udefn}
A \emph{simplicial $T$-complex} is a pair $(K,T)$, where $K$ is a [[simplicial set|simplicial set]] and $T = (T_n)_{n\geq 1}$ is a graded subset of $K$ with $T_n\subseteq  K_n$. Elements of $T$ are called \emph{[[thin]]}. The thin structure satisfies the following axioms:

\begin{itemize}%
\item Every degenerate element is thin.
\item Every [[horn]] in $K$ has a unique thin filler.
\item If all faces but one of a thin element are thin, then so also is the remaining face.

\end{itemize}
\end{udefn}
\begin{uex}
\begin{enumerate}%
\item A closely related idea is that of [[group T-complex]]. Group $T$-complexes form a category equivalent to reduced [[crossed complex]]es. Any group $T$-complex has an underlying simplicial set, which is a simplicial $T$-complex.


\item The nerve of a [[crossed complex]] has a natural T-complex structure. In a bit more detail, if $\mathsf{C}$ is a crossed complex, its nerve is given by $Ner(\mathsf{C})_n = Crs(\pi(n),\mathsf{C})$, where $\pi(n)$ is the free crossed complex on the $n$-simplex, $\Delta[n]$. This singular complex description shows that if we have an $n$-simplex $f : \pi(n) \to \mathsf{C}$, and declare it to be $\backslash$emph\{thin\} if the image $f(\iota_n)$ of the top dimensional generator in $\pi(n)$ is trivial, then the resulting collection of thin simplices determines a $T$- complex structure on the nerve.



\end{enumerate}
\end{uex}
\hypertarget{results}{}\subsubsection*{{Results}}\label{results}

\begin{itemize}%
\item Simplical $T$-complexes together with maps between them which preserve `thinness' form a category that is equivalent to that of [[crossed complexes]] and thus to strict $\infty$-groupoids. The uniqueness of the thin filler is exactly what gives a definite composition in the model.

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

\begin{itemize}%
\item [[T-complex]]

\begin{itemize}%
\item \textbf{simplicial $T$-complex}


\item [[cubical T-complex]]


\item [[group T-complex]]



\end{itemize}

\item [[algebraic Kan complex]]



\end{itemize}
Together with very similar ideas of [[John Roberts]], adapted by [[Ross Street]], the notion of $T$-complex is one of the precursors of [[Dominic Verity]]`s notion of [[complicial set|complicial set]].

They are also related to [[Jack Duskin]]`s notion of [[hypergroupoid]]. (The connection is explored in the papers by Nan Tie listed below.)

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

Relevant references for simplicial T-complexes include:

\begin{itemize}%
\item [[Ronnie Brown]], \emph{An Introduction to simplicial T-complexes}, \href{http://ehres.pagesperso-orange.fr/Cahiers/brownem32.pdf}{Esquisses Math. 32 (1983) Part 1}


\item M.K. Dakin, \emph{Kan complexes and multiple groupoid structures}, Ph.D Thesis, University of Wales, Bangor, 1977. \href{http://ehres.pagesperso-orange.fr/Cahiers/dakinEM.pdf}{Esquisses Math. (1983) 32 Part 2}


\item N. Ashley, \emph{Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan}, Ph.D Thesis University of Wales, Bangor, 1978; \href{https://eudml.org/doc/268359}{Dissertationes Math., 165, (1989), 11 -- 58}. \href{http://ehres.pagesperso-orange.fr/Cahiers/AshleyEM32.pdf}{Esquisses Math. (1983) 32 Part 3}


\item G. Nan Tie, \emph{A Dold-Kan theorem for crossed complexes}, J. Pure Appl. Alg., 56, (1989.), 177 --- 194.


\item G. Nan Tie, \emph{Iterated W and T-groupoids}, J. Pure Appl. Alg., 56, (1989), 195 --- 209.


\item [[Ronnie Brown]], and [[P.J. Higgins]], \emph{On the algebra of cubes} J. Pure Appl. Algebra 21 (1981) 233--260.


\item R. Brown, and P.J. Higgins, \emph{The equivalence of $\omega$-groupoids and cubical $T$-complexes} Cahiers Topologie G'eom. Diff'erentielle 22 (1981) 349--370.



\end{itemize}
[[!redirects simplical T-complex]] [[!redirects simplicial T-complexes]]



\end{document}