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\begin{document}

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\section*{cubical category}

The term \textbf{cubical category} has at least two common meanings. Thus, to avoid ambiguity, it is perhaps better to avoid it entirely and use an equivalent, unambiguous term for the particular meaning one has in mind.

\begin{itemize}%
\item The \textbf{cubical category} $\Box$ is the domain category for the presheaf category of [[cubical set|cubical sets]]. To avoid ambiguity, $\Box$ may also be called the \textbf{[[cube category]]} (which see, for its definition) or the \textbf{cubical indexing category}.


\item A \textbf{cubical category} is a [[cubical object]] in [[Cat]] (that is, a functor from $\Box^{op}$ to $Cat$), just like a cubical set is a cubical object in [[Set]], a cubical space is a cubical object in [[Top]], a cubical abelian group is a cubical object in [[Ab]], and so on. To avoid ambiguity, \textbf{[[cubical object in Cat|cubical objects in Cat]]} may be called exactly that.



\end{itemize}
A third use of the adjective ``cubical'' with a type of category is the following.

\begin{itemize}%
\item A \textbf{[[cubical ∞-category]]} is a [[cubical set]] equipped with operations giving a way to compose cubes with each other, in the same way that an ordinary [[∞-category]] is a [[globular set]] equipped with composition operations.

\end{itemize}


\end{document}