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\begin{document}

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\section*{semicategory}

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

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

[[!include category theory - 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToCategories}{Relation to categories}\dotfill \pageref*{RelationToCategories} \linebreak
\noindent\hyperlink{nerves_and_semisimplicial_sets}{Nerves and semi-simplicial sets}\dotfill \pageref*{nerves_and_semisimplicial_sets} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \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}

The notion of \emph{semicategory} or \emph{non-unital category} is like that of \emph{[[category]]} but omitting the requirement of [[identity]]-morphisms.

This generalizes the notions of [[semigroup]], [[semiring]], etc:

\begin{itemize}%
\item a [[semigroup]] is (the [[hom-set]] of) a semicategory with a single object;


\item a [[semiring]] is (the [[hom-set]] of) a semicategory [[enriched category|enriched]] in [[Ab]] with a single object.



\end{itemize}
Semicategories, like categories, appear as [[semipresheaf|semipresheaves]] on the category with two objects and two morphisms.

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

\begin{defn}
\label{SemiCategory}\hypertarget{SemiCategory}{}
A (small) \textbf{semicategory} or \textbf{non-unital category} $\mathcal{C}$ consists of

\begin{itemize}%
\item a [[set]] $\mathcal{C}_0$ of \emph{[[objects]]};


\item a set $\mathcal{C}_1$ of \emph{[[morphisms]]} (or \emph{arrows});


\item two [[functions]] $s, t : \mathcal{C}_1 \to \mathcal{C}_0$ called \emph{source} (or \emph{domain}) and \emph{target} (or \emph{codomain});

\begin{itemize}%
\item one writes $f : x \to y$ if $s(f) = x$ and $t(f) = y$;

\end{itemize}

\item a [[function]] $\circ \colon \mathcal{C}_1 \times_{t,s} \mathcal{C}_1 \to \mathcal{C}_1$ ([[composition]]) from the set of pairs of morphisms such that the target of the first is the source of the second;



\end{itemize}
such that the following properties are satisfied:

\begin{itemize}%
\item source and target are respected by composition: $s(g \circ f) = s(f)$ and $t(g\circ f) = t(g)$;


\item composition is \emph{[[associativity|associative]]}: $(h \circ g)\circ f = h\circ (g \circ f)$ whenever $t(f) = s(g)$ and $t(g) = s(h)$.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If one added to this definition the existence of a function $i \colon C_0 \to C_1$ such that for all $c \in C_0$ the morphism $i(c)$ is an [[identity]] on $c$ under the given [[composition]], then one has the defintion of a [[category]].

However, having [[identities]] is just an extra [[property]] on a semi-category, not extra [[structure]]. For more on this see below at \emph{\hyperlink{RelationToCategories}{Relation to categories}}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
One often writes $hom(x,y)$, $hom_C(x,y)$, or $C(x,y)$ for the collection of morphisms $f : x \to y$; the latter two have the advantage of making clear which category is being discussed. People also often write $x \in C$ instead of $x \in C_0$ as a short way to indicate that $x$ is an object of $C$. Also, some people write $Ob(C)$ and $Mor(C)$ instead of $C_0$ and $C_1$.

\end{remark}
\begin{defn}
\label{SemiFunctor}\hypertarget{SemiFunctor}{}
For $\mathcal{C}, \mathcal{D}$ two semicategories, a [[semi-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is a pair of [[functions]] $F_0 \colon \mathcal{C}_0 \to \mathcal{D}_0$, $F_1 \colon \mathcal{C}_1 \to \mathcal{D}_1$ that respects all the given [[structure]] in the obvious way.

Write $SemiCat$ for the ([[large category|large]]) [[category]] whose objects are semicategories, and whose morphisms are semifunctors.

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

\hypertarget{RelationToCategories}{}\subsubsection*{{Relation to categories}}\label{RelationToCategories}

We discuss the relation of semicategories to [[categories]]. (See for instance the beginning of (\hyperlink{Harpaz}{Harpaz}) for a quick review of basics, with an eye towards their generalization to the relation between [[complete Segal spaces]] and [[complete semi-Segal spaces]].)

\begin{defn}
\label{ForgetfulFromCatToSemicat}\hypertarget{ForgetfulFromCatToSemicat}{}
There is an evident [[forgetful functor]]

\begin{displaymath}
U \colon Cat \to SemiCat
\end{displaymath}
from the [[category]] [[Cat]] of [[categories]] to that of semicategories, def. \ref{SemiFunctor}, given simply by forgetting the identity-assigning map $i \colon \mathcal{C}_0 \to \mathcal{C}_1$ in a category.

\end{defn}
\begin{defn}
\label{SetNeutralElementsInEndoSemiMonoids}\hypertarget{SetNeutralElementsInEndoSemiMonoids}{}
For $\mathcal{C}$ a semi-category, def. \ref{SemiCategory}, write

\begin{displaymath}
Id(\mathcal{C}_1) \hookrightarrow \mathcal{C}_1
\end{displaymath}
for the [[subset]] on those morphisms which are [[endomorphisms]] on some object $x \in \mathcal{C}_0$ and such that they are neutral elements in their endomorphisms [[semimonoids]] $End_{\mathcal{C}}(x)$.

\end{defn}
\begin{prop}
\label{CategoriesAreSemicategoriesWithUnits}\hypertarget{CategoriesAreSemicategoriesWithUnits}{}
A semicategory is the semicategory underlying a category, hence is in the image of the functor $U$ of def. \ref{ForgetfulFromCatToSemicat}, precisely if every object has a neutral [[endomorphism]], hence precisely if the composite diagonal function in

\begin{displaymath}
\itexarray{
    Id(\mathcal{C}_1) &\hookrightarrow& \mathcal{C}_1
    \\
    & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{s}}
    \\
    && \mathcal{C}_0
  }
\end{displaymath}
is an [[isomorphism]], where the horizontal function is that of def. \ref{SetNeutralElementsInEndoSemiMonoids}.

Moreover, if a semicategory lifts to a category, it does so in a unique way: the functor $U \colon Cat \to SemiCat$ is an [[injection]] on [[isomorphism classes]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Equivalently one could use the target map instead of the source map in the formulation of prop. \ref{CategoriesAreSemicategoriesWithUnits}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The diagram appearing in prop. \ref{CategoriesAreSemicategoriesWithUnits} is a simple version of the [[univalence]] condition appearing in definition of a [[complete semi-Segal space]], a [[category object in an (infinity,1)-category|semi-category object in an (infinity,1)-category]]. See there for more on this.

\end{remark}
\begin{prop}
\label{idempotents}\hypertarget{idempotents}{}
The functor $U$ of def. \ref{ForgetfulFromCatToSemicat} has a [[left adjoint]], which [[free functor|freely]] adjoins identity morphisms to a semicategory in the obvious way. It also has a [[right adjoint]], which sends a semicategory $S$ to the category whose objects are the [[idempotents]] of $S$ and whose morphisms are the morphisms of $S$ that commute suitably with them, as described at [[Karoubi envelope]]. Indeed, the [[monad]] on [[Cat]] generated by this latter [[adjunction]] is exactly the monad for \emph{[[idempotent completion]]}, also called [[Cauchy completion]]. (Note, however, that this is not a [[2-monad]], because the right adjoint of $U$ is not a [[2-functor]].)

\end{prop}
\hypertarget{nerves_and_semisimplicial_sets}{}\subsubsection*{{Nerves and semi-simplicial sets}}\label{nerves_and_semisimplicial_sets}

The [[nerve]] of a semicategory is a [[semi-simplicial set]] which satisfies the [[Segal conditions]].

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

Start with the category of metric spaces and short maps. An occasionally useful semicategory can be formed from it by considering the nonempty spaces and strictly contractive functions.

This is a semicategory, since:

\begin{itemize}%
\item the composition of two strictly contractive functions is strictly contractive
\item identity maps are not contractive (they are trivial isometries)

\end{itemize}
The interest in this semicategory arises from the fact that all morphisms $f : A \to A$ have unique fixed points, by Banach's fixed point theorem.

\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

The concept of semicategory has more or less evident analogs and generalizations in [[higher category theory]].

For models of higher categories by [[simplicial set]]s, i.e. presehaves on the [[simplex category]] (such as [[Kan complex]]es, [[quasi-categories]], [[weak complicial set]]s) the corresponding semi-category notion is obtained by discarding the degeneracy maps (which are the identity-assigning maps in the simplicial framework), i.e. by considering just presheaves on the subcategory $\Delta_+ \subset \Delta$ on injective morphisms (see the discuss of $\Delta_+$ at [[Reedy model structure]] for more details).

Accordingly, there is the semi-category analog of a [[Segal space]], called a \emph{[[semi-Segal space]]}.

[[Simpson's conjecture]] says that every $\infty$-category has a model where all [[composition]] operations are strict and only the [[unit law]]s hold up to [[coherent]] homotopies. This would mean that the $\infty$-semicategory underlying any $\infty$-category can always be chosen to be strict.

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

\begin{itemize}%
\item [[semipresheaf]]


\item [[regular semicategory]]


\item [[non-unital ring]]


\item [[semi-simplicial set]]


\item [[semi-Segal space]]



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

Enriched semicategory theory is developed in

\begin{itemize}%
\item M.-A. Moens, U. Bernani-Canani, [[Francis Borceux|F. Borceux]], \emph{On regular presheaves and regular semi-categories} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XLIII} no.3 (2002) pp.163-190. (\href{http://www.numdam.org/item?id=CTGDC_2002__43_3_163_0}{numdam})

\end{itemize}
This is turned one notch further in

\begin{itemize}%
\item [[Isar Stubbe]], \emph{Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XLVI} no.2 (2005) pp.99-121. (\href{http://www.numdam.org/item/CTGDC_2005__46_2_99_0}{numdam})

\end{itemize}
Semicategories and semigroups are mentioned in section 2 in

\begin{itemize}%
\item W. Dale Garraway, \emph{Sheaves for an involutive quantaloid}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 46 no. 4 (2005), p. 243-274 (\href{http://www.numdam.org/item?id=CTGDC_2005__46_4_243_0}{numdam})

\end{itemize}
Semicategories with an eye towards their generalization to [[semi-Segal spaces]] are briefly discussed at the beginning of

\begin{itemize}%
\item [[Yonatan Harpaz]], \emph{Quasi-unital $\infty$-Categories} (\href{http://arxiv.org/abs/1210.0212}{arXiv:1210.0212})

\end{itemize}
Structures obtained by further relaxing also the [[associativity]] law are discussed in

\begin{itemize}%
\item Salvatore Tringali, \emph{Plots and Their Applications - Part I: Foundations} (\href{http://arxiv.org/abs/1311.3524}{arXiv:1311.3524})

\end{itemize}
[[!redirects semicategories]]

[[!redirects semi-category]] [[!redirects semi-categories]]

[[!redirects category without units]] [[!redirects categories without units]]



\end{document}