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\begin{document}

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\section*{homotopy (as an operation)}

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

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

[[!include homotopy - contents]]

\begin{itemize}%
\item This page is about homotopy as an operation. For homotopy as a transformation, see [[homotopy]].

\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{homotopy_sets_in_homotopy_categories}{}\section*{{Homotopy sets in homotopy categories}}\label{homotopy_sets_in_homotopy_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Ordinary \emph{homotopy} is a way to probe objects in an [[(∞,1)-topos]] $\mathbf{H}$ by mapping [[spheres]] into them:

the ordinary [[homotopy group]] $\pi_n(X,x)$ of an object $X \in \mathbf{H}$ is the fiber over $x \in X$ of the morphism

\begin{displaymath}
[S^n, X]_{\mathbf{H}} \to \tau_0 X \simeq \pi_0(X)
\end{displaymath}
In this sense \emph{homotopy} is the notion that is [[Eckmann-Hilton duality|Eckmann-Hilton dual]] to [[cohomology]].

For a detailed discussion see

\begin{itemize}%
\item [[homotopy groups in an (∞,1)-topos]]

\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
This duality suggests that more generally we may be entitled to speak for $B$ and $X$ objects in $\mathbf{H}$ of

\begin{displaymath}
H(B,X) := \pi_0 \mathbf{H}(B,X)_*
\end{displaymath}
as the \textbf{homotopy of $X$ with co-coefficients in $B$}.

Examples of such constructions exist, but are rarely thought of (or even recognized as) generalizations of the notion of homotopy. Rather, by the above duality, the same situation is usually regarded in the context of [[cohomology]], which, still by the above duality, is just as well.

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

\begin{itemize}%
\item [[group homotopy]]


\item [[generalized (Eilenberg-Steenrod) homotopy]]


\item abelian cosheaf homotopy


\item nonabelian homotopy


\item [[?ech homotopy]]


\item differential homotopy


\item twisted homotopy



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

[[!include homotopy-homology-cohomology]]

\begin{itemize}%
\item [[homotopy]], [[homotopy group]],


\item [[homotopy equivalence]], [[weak homotopy equivalence]]


\item [[homotopy groups in an (∞,1)-topos]] .



\end{itemize}
[[!redirects homotopy as an operation]] [[!redirects homotopy (as an operation)]]



\end{document}