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\begin{document}

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\section*{cartesian multicategory}

\hypertarget{cartesian_multicategories}{}\section*{{Cartesian multicategories}}\label{cartesian_multicategories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{explicit}{Explicit}\dotfill \pageref*{explicit} \linebreak
\noindent\hyperlink{as_a_generalized_multicategory}{As a generalized multicategory}\dotfill \pageref*{as_a_generalized_multicategory} \linebreak
\noindent\hyperlink{as_a_free_category_with_finite_products}{As a free category with finite products}\dotfill \pageref*{as_a_free_category_with_finite_products} \linebreak
\noindent\hyperlink{representability}{Representability}\dotfill \pageref*{representability} \linebreak
\noindent\hyperlink{syntax_semantics_and_type_theory}{Syntax, semantics, and type theory}\dotfill \pageref*{syntax_semantics_and_type_theory} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak


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

A \textbf{cartesian multicategory} is a [[multicategory]]-like structure which is related to categories with finite [[products]] in the same way that ordinary multicategories are related to [[monoidal categories]].

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

\hypertarget{explicit}{}\subsubsection*{{Explicit}}\label{explicit}

A \textbf{cartesian multicategory} is a symmetric [[multicategory]] equipped with:

\begin{itemize}%
\item \emph{duplication} or \emph{diagonal} or \emph{contraction} operations

\begin{displaymath}
hom(c_0,\dots,c_{k-1},c_k,c_k,c_{k+1},\dots,c_{n-1} \;;\; d) \to hom(c_0,\dots,c_{k-1},c_k,c_{k+1},\dots,c_{n-1} \;;\; d)
\end{displaymath}

\item \emph{deletion} or \emph{projection} or \emph{weakening} operations

\begin{displaymath}
hom(c_0,\dots,c_{k-1},c_{k+1},\dots,c_{n-1} \;;\; d) \to hom(c_0,\dots,c_{k-1},c_k,c_{k+1},\dots,c_{n-1} \;;\; d)
\end{displaymath}


\end{itemize}
which satisfy certain evident axioms.

\hypertarget{as_a_generalized_multicategory}{}\subsubsection*{{As a generalized multicategory}}\label{as_a_generalized_multicategory}

A cartesian multicategory can equivalently be defined as a [[generalized multicategory]] relative to the monad on [[Cat]] (or more precisely [[Prof]]) whose algebras are categories with (strict) [[finite products]].

\hypertarget{as_a_free_category_with_finite_products}{}\subsubsection*{{As a free category with finite products}}\label{as_a_free_category_with_finite_products}

A cartesian multicategory can also be defined as a category with specified [[finite products]] whose set of objects under the ``product'' operation is a free monoid on specified generators.

When there is exactly one such generator, this recovers the definition of a [[Lawvere theory]]; thus a cartesian multicategory may be considered a ``colored'' or ``many-object'' Lawvere theory. Note, though, that the morphisms of cartesian multicategories are more restrictive than morphisms of finite-product categories; they are required to take generators to generators.

\hypertarget{representability}{}\subsection*{{Representability}}\label{representability}

A cartesian multicategory, like an ordinary multicategory, is \textbf{[[representable multicategory|representable]]} if for any finite list $(c_1,\dots,c_n)$ of objects there exists an object ``$c_1 \times \cdots \times c_n$'' and a morphism $c_1,\dots,c_n \to c_1 \times \cdots \times c_n$ which is universal, in that the following induced functions are all bijections:

\begin{displaymath}
hom(d_1,\dots,d_k, c_1 \times \cdots \times c_n , e_1,\dots,e_m \;;\; f) \to
hom(d_1,\dots,d_k, c_1,\dots,c_n, e_1,\dots,e_m \;;\; f)
\end{displaymath}
Just as representable (symmetric) multicategories are equivalent to (symmetric) monoidal categories, representable cartesian multicategories are equivalent to [[cartesian monoidal categories]].

More abstractly, representable cartesian multicategories are the algebras for a [[colax-idempotent 2-monad]] on the 2-category of cartesian multicategories, whose algebras are categories with finite products.

\hypertarget{syntax_semantics_and_type_theory}{}\subsection*{{Syntax, semantics, and type theory}}\label{syntax_semantics_and_type_theory}

Small cartesian multicategories may be considered presentations of ``many-sorted finite-product theories''. On the spectrum between syntax and semantics, they sit in between a fully syntactic presentation (such as a [[type theory]]) and a fully incarnated semantic one (the [[walking structure|free category]] with finite products generated by a model of some theory).

Cartesian multicategories are also a natural place to talk about the semantics of [[type theory]]. Every type theory (with [[contraction rule|contraction]] and [[weakening rule|weakening]], and without [[dependent types]]) gives rise to a cartesian multicategory whose objects are its types and whose multimorphisms are its terms. This is in contrast to the usual construction of an ordinary category from a type theory, in which we have to take the objects to be [[contexts]] in order to recapture all the terms. This usual [[category of contexts]] can be recaptured as the free category-with-finite-products generated by this cartesian multicategory of types. Similarly, we can talk about models of such a type theory in any cartesian multicategory.

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[multicategory]]
\item [[generalized multicategory]]
\item [[Lawvere theory]]
\item A cartesian multicategory with one object is called a [[clone]].

\end{itemize}
[[!redirects cartesian multicategory]] [[!redirects cartesian multicategories]] [[!redirects finite-product multicategory]] [[!redirects finite-product multicategories]]



\end{document}