\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*{semicartesian monoidal category}

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

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{semicartesian_monoidal_categories}{}\section*{{Semicartesian monoidal categories}}\label{semicartesian_monoidal_categories}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{internal_logic}{Internal logic}\dotfill \pageref*{internal_logic} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{semicartesian_vs_cartesian}{Semicartesian vs. cartesian}\dotfill \pageref*{semicartesian_vs_cartesian} \linebreak
\noindent\hyperlink{definition_in_terms_of_projections}{Definition in terms of projections}\dotfill \pageref*{definition_in_terms_of_projections} \linebreak
\noindent\hyperlink{colax_functors}{Colax functors}\dotfill \pageref*{colax_functors} \linebreak
\noindent\hyperlink{semicartesian_operads}{Semicartesian operads}\dotfill \pageref*{semicartesian_operads} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak


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

A [[monoidal category]] is \textbf{semicartesian} if the unit for the tensor product is a [[terminal object]], a weakening of the concept of [[cartesian monoidal category]].

Many semicartesian monoidal categories are also [[symmetric monoidal category|symmetric]], and sometimes that is included in the definition.

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

Some examples of semicartesian monoidal categories that are not cartesian include the following.

\begin{itemize}%
\item The category of [[Poisson manifold]]s with the usual product of Poisson manifolds as its tensor product.


\item The opposite of the category of [[associative algebra]]s over a given [[base field]] $k$ with its usual tensor product $A \otimes B$.


\item The category [[Cat]] with the non-standard non-cartesian [[funny tensor product]].


\item The category of [[strict 2-categories]] with the [[Gray tensor product]], and the category of [[strict omega-categories]] with the [[Crans-Gray tensor product]].


\item The category of [[affine space|affine spaces]] made into a [[closed monoidal category]] where the [[internal hom]] has $hom(x,y)$ being the set of affine linear maps from $x$ to $y$, made into an affine space via pointwise operations.


\item The category of [[convex space|convex spaces]], also known as `barycentric algebras', made into a closed monoidal category where the internal hom has $hom(x,y)$ being the set of convex linear maps from $x$ to $y$, made into an barycentric algebra via pointwise operations.


\item The thin category associated to the linear order $([0, \infty], \ge)$ of extended nonnegative real numbers with addition as the tensor product and [[internal hom]] as truncated subtraction.


\item If $(M, \otimes, I)$ is any monoidal category, $I$ being the monoidal unit, the [[slice category]] $M/I$ inherits a monoidal product given by

\begin{displaymath}
(X \stackrel{f}{\to} I) \otimes (Y \stackrel{g}{\to} I) = (X \otimes Y \stackrel{f \otimes g}{\to} I \otimes I \cong I)
\end{displaymath}
where the isomorphism displayed is the canonical one. This monoidal product is semicartesian. The forgetful functor $\Sigma: M/I \to M$ is strong monoidal, and is universal in the sense of exhibiting the fact that semicartesian monoidal functors and strong monoidal functors form a coreflective sub-bicategory of the bicategory of monoidal categories and strong monoidal functors. (Check this.)


\item The category of [[nominal sets]] is cartesian closed but also semi-cartesian closed if taken with the separated product, see Chapter 3.4 of Pitts' monograph Nominal Sets.



\end{itemize}
\hypertarget{internal_logic}{}\subsection*{{Internal logic}}\label{internal_logic}

The [[internal logic]] of a (symmetric) semicartesian monoidal category is [[affine logic]], which is like [[linear logic]] but permits the [[weakening rule]] (and also the [[exchange rule]], if the monoidal structure is symmetric).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{semicartesian_vs_cartesian}{}\subsubsection*{{Semicartesian vs. cartesian}}\label{semicartesian_vs_cartesian}

In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms

\begin{displaymath}
p_x : x \otimes y  \to x
\end{displaymath}
\begin{displaymath}
p_y : x \otimes y \to y
\end{displaymath}
given by

\begin{displaymath}
x \otimes y \stackrel{1 \otimes e_y}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x
\end{displaymath}
and

\begin{displaymath}
x \otimes y \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes y \stackrel{\ell_y}{\longrightarrow} y
\end{displaymath}
respectively, where $e$ stands for the unique morphism to the terminal object and $r$, $\ell$ are the right and left unitors. We can thus ask whether $p_x$ and $p_y$ make $x \otimes y$ into the [[product]] of $x$ and $y$. If so, it is a theorem that $C$ is a cartesian monoidal category. (This theorem has been observed by Eilenberg and Kelly (\hyperlink{EilKel66}{1966}, p.551), but they may not have been the first to note it.)

Alternatively, suppose that $(C, \otimes, I)$ is a monoidal category equipped with monoidal natural transformations $e_x : x \to I$ and $\Delta_x: x \to x \otimes x$ such that

\begin{displaymath}
x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes x \stackrel{\ell_x}{\longrightarrow} x
\end{displaymath}
and

\begin{displaymath}
x \stackrel{\Delta_x}{\longrightarrow} x \otimes x \stackrel{1 \otimes e_x}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x
\end{displaymath}
are identity morphisms. Then $(C, \otimes, I)$ is a cartesian monoidal category.

So, suppose $(C, \otimes, 1)$ is a semicartesian monoidal category. The unique map $e_x : x \to I$ is a monoidal natural transformation. Thus, if there exists a monoidal natural transformation $\Delta_x: x \to x \otimes x$ obeying the above two conditions, $(C, \otimes, 1)$ is cartesian. The converse is also true.

The characterization of cartesian monoidal categories in terms of $e$ and $\Delta$, apparently discovered by Robin Houston, is mentioned here:

\begin{itemize}%
\item John Baez, Universal algebra and diagrammatic reasoning, 2006. \href{http://math.ucr.edu/home/baez/universal/}{pdf}

\end{itemize}
and as of 2014, Nick Gurski plans to write up the proof in a paper on semicartesian monads.

\hypertarget{definition_in_terms_of_projections}{}\subsubsection*{{Definition in terms of projections}}\label{definition_in_terms_of_projections}

For a monoidal category to be semicartesian it suffices that it admit a family of ``projection morphisms''. Specifically, suppose $C$ is a monoidal category together with natural ``projection'' transformations $\pi^1_{X,Y}:X\otimes Y \to X$ such that $\pi^1_{I,I}:I\otimes I\to I$ is the unitor isomorphism. Then the composites $Y \cong I\otimes Y \to I$ form a cone under the identity functor with vertex $I$ whose component at $I$ is the identity; hence $I$ is a [[terminal object]] and so $C$ is semicartesian.

However, it doesn't follow from this that the given projections $\pi^1_{X,Y}$ are the same as those derivable from semicartesianness! For that one needs extra axioms; see \href{https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html#c056710}{this cafe discussion} for details.

\hypertarget{colax_functors}{}\subsubsection*{{Colax functors}}\label{colax_functors}

It is well-known that any functor between cartesian monoidal categories is automatically and uniquely [[colax monoidal functor|colax monoidal]]; the colax structure maps are the comparison maps $F(x\times y) \to F x \times F y$ for the cartesian product. (This also follows from abstract nonsense given that the [[2-monad]] for cartesian monoidal categories is [[colax-idempotent 2-monad|colax-idempotent]].) An inspection of the proof reveals that this property only requires the domain category to be semicartesian monoidal, although the codomain must still be cartesian.

\hypertarget{semicartesian_operads}{}\subsubsection*{{Semicartesian operads}}\label{semicartesian_operads}

The notion of [[semicartesian operad]] is a type of [[generalized multicategory]] which corresponds to semicartesian monoidal categories in the same way that [[operads]] correspond to (perhaps symmetric) monoidal categories and [[Lawvere theories]] correspond to cartesian monoidal categories. Applications of semicartesian operads include:

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2009/10/generalized_operads_in_classic.html}{Generalized operads in classical algebraic topology} (blog post) -- this also uses the above fact about colax functors


\item \href{http://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html#c038131}{Characterizing finite measure spaces} (blog comment)



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

\begin{itemize}%
\item [[cartesian monoidal category]]


\item A [[relevance monoidal category]] is the ``dual'' of a semicartesian monoidal category, with diagonals but not projections.



\end{itemize}
\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

\begin{itemize}%
\item [[Samuel Eilenberg|S. Eilenberg]], [[Max Kelly|M. G. Kelly]], \emph{Closed Categories} , pp.421-562 in Eilenberg et al. (eds.), \emph{Proceedings of the Conference on Categorical Algebra - La Jolla 1965} , Springer Heidelberg 1966.


\item \href{https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html}{Monoidal Categories with Projections} (blog discussion)



\end{itemize}
[[!redirects semicartesian category]] [[!redirects semi-cartesian category]] [[!redirects semi-Cartesian category]] [[!redirects semicartesian monoidal categories]] [[!redirects semi-cartesian monoidal category]] [[!redirects semi-cartesian monoidal categories]] [[!redirects semicartesian categories]] [[!redirects semi-cartesian categories]] [[!redirects semi-Cartesian categories]]



\end{document}