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\begin{document}

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\section*{Ab-enriched category}

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

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{enriched_categories}{}\section*{{$Ab$-enriched categories}}\label{enriched_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{finite_products_are_absolute}{Finite products are absolute}\dotfill \pageref*{finite_products_are_absolute} \linebreak
\noindent\hyperlink{zero_objects}{Zero objects}\dotfill \pageref*{zero_objects} \linebreak
\noindent\hyperlink{biproducts}{Biproducts}\dotfill \pageref*{biproducts} \linebreak
\noindent\hyperlink{as_a_generalisation_of_rings}{As a generalisation of rings}\dotfill \pageref*{as_a_generalisation_of_rings} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

An \emph{$Ab$-enriched category} (or, if small, \emph{ringoid}) is a [[category enriched]] over the [[monoidal category]] [[Ab]] of [[abelian groups]] with its usual [[tensor product]].

Sometimes they are called [[pre-additive category|pre-additive categories]], but sometimes that term also implies the existence of a [[zero object]].

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

Explicitly, an \textbf{$Ab$-enriched category} is a [[category]] $C$ such that for all objects $a,b$ the [[hom-set]] $Hom_C(a,b)$ is equipped with the structure of an [[abelian group]]; and such that for all triples $a,b,c$ of objects the [[composition]] operation $\circ_{a,b,c} : Hom_C(a,b) \times Hom_C(b,c) \to Hom_C(a,c)$ is bilinear. A \textbf{ringoid} is small $Ab$-enriched category.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item $Ab$-enriched categories are called ringoids since the concept is a [[horizontal categorification]] (or `oidification') of the concept of a [[ring]].


\item There is a canonical forgetful functor $Ab \to Set_*$ from abelian groups to [[pointed set]]s, which sends each group to its underlying set with point being the neutral element. Using this functor, every $Ab$-enriched category $C$ is in particular also a category that is enriched over pointed sets (that is, a category with [[zero morphisms]]). This is sufficient for there to be a notion of [[kernel]] and [[cokernel]] in $C$.


\item In general, [[abelian category|abelian categories]] are the most important examples of $Ab$-enriched categories. See [[additive and abelian categories]].



\end{itemize}
\hypertarget{finite_products_are_absolute}{}\subsection*{{Finite products are absolute}}\label{finite_products_are_absolute}

One of the remarkable facts about $Ab$-enriched categories is that finite [[products]] (and [[coproducts]]) are [[absolute limit|absolute limits]]. This implies that finite products coincide with finite coproducts, and are preserved by \emph{any} $Ab$-enriched functor.

\hypertarget{zero_objects}{}\subsubsection*{{Zero objects}}\label{zero_objects}

In an $Ab$-enriched category $C$, any [[initial object]] is also a [[terminal object]], hence a [[zero object]], and dually. An object $a\in C$ is a zero object just when its identity $1_a$ is equal to the zero morphism $0:a\to a$ (that is, the identity element of the abelian group $\hom_C(a,a)$). Expressed in this way, it is easy to see that any $Ab$-enriched functor preserves zero objects.

\hypertarget{biproducts}{}\subsubsection*{{Biproducts}}\label{biproducts}

For $c_1, c_2 \in C$ two objects in an $Ab$-enriched category $C$, [[generalized the|the]] [[product]] $c_1 \times c_2$ coincides with [[generalized the|the]] [[coproduct]] $c_1 \sqcup c_2$ when either exists. More precisely, when both exist, the canonical morphism

\begin{displaymath}
r : c_1 \sqcup c_2 \to c_1 \times c_2
\end{displaymath}
defined by

\begin{displaymath}
\left(
    c_i \to c_1 \sqcup c_2 \stackrel{r}{\to} c_1 \times c_2 \to c_j
  \right)
  =
  \left\{
    \itexarray{
      Id_c_i & if i = j
      \\
      0 & if i \neq j
    }
  \right.
  \,,
\end{displaymath}
which exists whenever $c_1\sqcup c_2$ and $c_1\times c_2$ do, is an [[isomorphism]]. This object is called a [[biproduct]] or (sometimes) a [[direct sum]] and is generally denoted

\begin{displaymath}
c_1 \oplus c_2.
\end{displaymath}
It can be characterized diagrammatically as an object $c_1\oplus c_2$ equipped with morphisms $q_i:c_i\to c_1\oplus c_2$ and $p_i:c_1\oplus c_2 \to c_i$ such that $p_i q_j = \delta_{i j}$ and $q_1 p_1 + q_2 p_2 = 1_{c_1\oplus c_2}$. Expressed in this form, it is clear that any $Ab$-enriched functor preserves biproducts.

\hypertarget{as_a_generalisation_of_rings}{}\subsection*{{As a generalisation of rings}}\label{as_a_generalisation_of_rings}

When using the term `ringoid', one often assumes a ringoid to be [[small category|small]].

Ringoids share many of the properties of (noncommutative) [[rings]]. For instance, we can talk about (left and right) [[modules]] over a ringoid $R$, which can be defined as $Ab$-enriched [[functors]] $R\to Ab$ and $R^{op}\to Ab$. [[bimodule|Bimodules]] over ringoids have a tensor product (the enriched [[tensor product of functors]]) under which they form a [[bicategory]], also known as the bicategory $Ab Prof$ of $Ab$-enriched [[profunctors]]. Modules over a ringoid also form an [[abelian category]] and thus have a [[derived category]].

One interesting operation on ringoids is the ($Ab$-enriched) [[Cauchy completion]], which is the completion under finite [[direct sums]] and [[split idempotents]]. In particular, the Cauchy completion of a ring $R$ is the category of [[finitely generated object|finitely generated]] [[projective object|projective]] $R$-modules (aka [[split monomorphism|split]] [[subobjects]] of finite-rank [[free object|free]] modules). Every ringoid is [[equivalence|equivalent]] to its Cauchy completion in the bicategory $Ab Prof$, and two ringoids are equivalent in $Ab Prof$ if and only if their Cauchy completions are [[equivalence of categories|equivalent]] as $Ab$-enriched categories. This sort of equivalence is naturally called [[Morita equivalence]].

See also [[dg-category]].

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

\begin{itemize}%
\item The category [[Ab]] is [[closed monoidal category|closed monoidal]] and hence canonically enriched over itself.


\item An $Ab$-enriched category with one object is precisely a [[ring]].


\item For any small $Ab$-enriched category $R$, the enriched [[presheaf category]] $[R^{op},Ab]$ is, of course, $Ab$-enriched. If $R$ is a ring, as above, then $[R^{op},Ab]$ is the category of $R$-modules.



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

\begin{itemize}%
\item [[hom-group]]


\item [[additive category]]



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

\begin{itemize}%
\item John Baez, \href{http://golem.ph.utexas.edu/category/2006/09/ringoids.html}{Ringoids}, blog
\item C. Weibel, [[An Introduction to Homological Algebra]], Cambridge Univ. Press
\item [[Daniel Murfet]], \emph{Localisation of ringoids}, \href{http://therisingsea.org/notes/LocalisationOfRingoids.pdf}{pdf} 2006 notes
\item N. Popescu, \emph{Abelian categories with applications to rings and modules}, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

\end{itemize}
[[!redirects Ab-enriched category]] [[!redirects Ab-enriched categories]] [[!redirects Ab-category]] [[!redirects Ab-categories]] [[!redirects ringoid]] [[!redirects ringoids]]



\end{document}