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\begin{document}

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\section*{2-rig}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{rigs}{}\section*{{$2$-rigs}}\label{rigs}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definitions}{Definitions}\dotfill \pageref*{Definitions} \linebreak
\noindent\hyperlink{EnrichedMonoidalCategories}{Enriched monoidal categories}\dotfill \pageref*{EnrichedMonoidalCategories} \linebreak
\noindent\hyperlink{MonoidalCompleteCateories}{Compatibly monoidal cocomplete categories}\dotfill \pageref*{MonoidalCompleteCateories} \linebreak
\noindent\hyperlink{CompatiblyMonoidalPresentableCategories}{Compatibly monoidal presentable categories}\dotfill \pageref*{CompatiblyMonoidalPresentableCategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{PropertiesInitialObject}{Initial object}\dotfill \pageref*{PropertiesInitialObject} \linebreak
\noindent\hyperlink{TannakaDuality}{Tannaka duality}\dotfill \pageref*{TannakaDuality} \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{$2$-rig} is supposed to be a [[categorification]] of that of a [[rig]]. Several inequivalent formalizations of this idea are in the literature.

Just as a rig is a multiplicative [[monoid]] whose underlying set also has a notion of addition, so a $2$-rig is a [[monoidal category]] whose underlying category also has a notion of addition, and we can describe this notion of addition in a few different ways.

Note that we don't expect a $2$-rig to have additive inverses; by the same argument as in the [[Eilenberg swindle]], they are unreasonable to expect. However, in a monoidal [[abelian category]], we have as close to additive inverses as is reasonable and so a categorification of a [[ring]].

Compare also the notion of [[rig category]].

\hypertarget{Definitions}{}\subsection*{{Definitions}}\label{Definitions}

Since [[categorification]] involves some arbitrary choices that will be determined by the precise intended application, there is a bit of flexibility of what exactly one may want to call a \emph{2-ring}. We first list some immediate possibilities of classes of monoidal and enriched categories that one may want to think of as 2-rings:

\begin{itemize}%
\item \hyperlink{EnrichedMonoidalCategories}{Enriched monoidal categories}

\end{itemize}
But a central aspect of an ordinary ring is the [[distributivity law]] which says that the product in the ring preserves sums. Since sums in a 2-ring are given by [[colimits]], this suggests that a 2-ring should be a cocomplete category which is compatibly [[monoidal category|monoidal]] in that the the tensor product preserves colimits:

\begin{itemize}%
\item \hyperlink{MonoidalCompleteCateories}{Compatibly monoidal cocomplete categories}

\end{itemize}
But there are still more properties which one may want to enforce, notably that homomorphisms of 2-rings form a [[2-abelian group]]. This is achieved by demanding the underlying category to be not just cocomplete by [[presentable category|presentable]]:

\begin{itemize}%
\item \hyperlink{CompatiblyMonoidalPresentableCategories}{Compatibly monoidal presentable categories}.

\end{itemize}
\hypertarget{EnrichedMonoidalCategories}{}\subsubsection*{{Enriched monoidal categories}}\label{EnrichedMonoidalCategories}

\begin{enumerate}%
\item A \textbf{$2$-rig} might be an [[Ab-enriched category]] which is [[enriched monoidal category|enriched monoidal]].


\item A \textbf{$2$-rig} might be an [[additive category]] which is enriched monoidal.


\item A \textbf{$2$-rig} might be a [[distributive monoidal category]]: a monoidal category with finite [[coproducts]] such that the monoidal product distributes over the coproducts.


\item A \textbf{$2$-rig} might be a [[closed monoidal category]] with finite coproducts.


\item Finally, a \textbf{$2$-ring} is a monoidal [[abelian category]].



\end{enumerate}
Note that (2) is a special case of both (1) and (3), which are independent. (4) is a special case of (3), by the [[adjoint functor theorem]]. (5) is a special case of (2), of course.

\hypertarget{MonoidalCompleteCateories}{}\subsubsection*{{Compatibly monoidal cocomplete categories}}\label{MonoidalCompleteCateories}

In (\hyperlink{BaezDolan}{Baez-Dolan}) the following is considered:

\begin{defn}
\label{BD2Rig}\hypertarget{BD2Rig}{}
A \emph{2-rig} is a [[monoidal category|monoidal]] [[colimit|cocomplete category]] where the [[tensor product]] respects [[colimits]].

\end{defn}
One can define [[braided monoidal category|braided]] and [[symmetric monoidal category|symmetric]] 2-rigs in this sense (and indeed, also in the other senses listed above). In particular, there is a [[2-category]] $\mathbf{Symm2Rig}$ with:

\begin{itemize}%
\item symmetric monoidal cocomplete categories where the monoidal product distributes over colimits as objects,


\item symmetric monoidal cocontinuous functors as 1-morphisms,


\item symmetric monoidal natural transformations as 2-morphisms.



\end{itemize}
\hypertarget{CompatiblyMonoidalPresentableCategories}{}\subsubsection*{{Compatibly monoidal presentable categories}}\label{CompatiblyMonoidalPresentableCategories}

The following refines the \hyperlink{MonoidalCompleteCateories}{above} by demanding the underlying category of a 2-ring to be not just cocomplete but even a [[presentable category]]. This was motivated in (\hyperlink{CJF}{CJF, remark 2.1.10}).

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
2 Ab \in 2Cat
\end{displaymath}
for the [[2-category]] of [[presentable categories]] and [[colimit]]-preserving [[functors]] between them.

\end{defn}
(\hyperlink{CJF}{CJF, def. 2.1.8})

\begin{remark}
\label{}\hypertarget{}{}
By the [[adjoint functor theorem]] this is equivalently the 2-category of presentable categories and [[left adjoint]] functors between them.

\end{remark}
\begin{example}
\label{CategoryOfModulesAs2AbelianGroup}\hypertarget{CategoryOfModulesAs2AbelianGroup}{}
Given an ordinary [[ring]] $R$, its [[category of modules]] $Mod_R$ is presentable, hence may be regarded as a 2-abelian group.

\end{example}
(\hyperlink{CJF}{CJF, example 2.1.5})

\begin{prop}
\label{}\hypertarget{}{}
The 2-category $2Ab$ is a [[closed monoidal 2-category|closed]] [[symmetric monoidal 2-category]] with respect to the [[tensor product]] $\boxtimes \colon 2Ab \times 2Ab \to 2Ab$ such that for $A,B, C \in 2Ab$, $Hom_{2Ab}(A \boxtimes B, C)$ is equivalently the full [[subcategory]] of [[functor category]] $Hom_{Cat}(A \times B, C)$ on those that are [[bilinear function|bilinear]] in that they preserve [[colimits]] in each argument separately.

\end{prop}
See also at [[Pr(∞,1)Cat]] for more on this.

\begin{example}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[small category]], the [[category of presheaves]] $Set^{\mathcal{C}}$ is [[presentable category|presentable]] and

\begin{displaymath}
Set^{\mathcal{C}_1} \boxtimes
  Set^{\mathcal{C}_2}
  \simeq
  Set^{\mathcal{C}_1 \times \mathcal{C}_2}
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{TensorProductModuleCategoriesAsOf2AbelianGroups}\hypertarget{TensorProductModuleCategoriesAsOf2AbelianGroups}{}
For $R$ a [[ring]] the [[category of modules]] $Mod_R$ is presentable and

\begin{displaymath}
Mod_{R_1} \boxtimes Mod_{R_2} 
  \simeq
  Mod_{R_1 \otimes R_2}
  \,,
\end{displaymath}
\end{example}
(\hyperlink{CJF}{CJF, example 2.2.7})

\begin{prop}
\label{EilenbergWattsTheorem}\hypertarget{EilenbergWattsTheorem}{}
For $R_1, R_2$ two rings, the category of 2-abelian group homomorphisms between the [[categories of modules]] is [[natural equivalence|naturally equivalent]] to that of $R_1$-$R_2$-[[bimodules]] and their [[intertwiners]]:

\begin{displaymath}
(-)\otimes (-)
  \;\colon\;
  {}_{R_1}Mod_{R_2}
  \stackrel{\simeq}{\to}  
  Hom_{2Ab}(Mod_{R_1}, Mod_{R_2})
  \,.
\end{displaymath}
The equivalence sends a bimodule $N$ to the functor given by the [[tensor product]] over $R_1$:

\begin{displaymath}
(-) \otimes N \;\colon\;
  Mod_{R_1} \to Mod_{R_2}
  \,.
\end{displaymath}
\end{prop}
This is the [[Eilenberg-Watts theorem]].

\begin{defn}
\label{2RingAsCompatiblyMonoidalPresentableCategory}\hypertarget{2RingAsCompatiblyMonoidalPresentableCategory}{}
Write

\begin{displaymath}
2Ring \in 2Cat
\end{displaymath}
for the [[2-category]] of [[monoid objects]] [[internalization|internal]] to $2 Ab$. An [[object]] of this 2-category we call a \textbf{2-ring}.

Equivalently, a 2-ring in this sense is a [[presentable category]] equipped with the structure of a [[monoidal category]] where the [[tensor product]] preserves [[colimits]].

\end{defn}
(\hyperlink{CJF}{CJF, def. 2.1.8})

\begin{example}
\label{}\hypertarget{}{}
The category [[Set]] with its [[cartesian product]] is a 2-ring and it is the [[initial object]] in $2Ring$.

\end{example}
(\hyperlink{CJF}{CJF, example 2.3.4})

\begin{example}
\label{}\hypertarget{}{}
The category [[Ab]] of [[abelian groups]] with its standard [[tensor product of abelian groups]] is a 2-ring.

\end{example}
\begin{example}
\label{CommutativeRingGivesCommutative2Ring}\hypertarget{CommutativeRingGivesCommutative2Ring}{}
For $R$ an ordinary [[commutative ring]], $Mod_R$ equipped with its usual [[tensor product of modules]] is a commutative 2-ring.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $R$ an ordinary [[ring]] and $Mod_R$ its ordinary [[category of modules]], regarded as a 2-abelian group by example \ref{CategoryOfModulesAs2AbelianGroup}, the structure of a 2-ring on $Mod_R$ is equivalently the structure of a [[sesquiunital sesquialgebra]] on $R$.

If $R$ is in addition a [[commutative ring]] that $Mod_R$ is a commutative 2-ring and is canonically an $Ab$-[[2-algebra]] in that

\begin{displaymath}
Ab \simeq Mod_{\mathbb{Z}} \to Mod_R
  \,.
\end{displaymath}
\end{example}
(\hyperlink{CJF}{CJF, example 2.3.7})

\begin{defn}
\label{}\hypertarget{}{}
For $A$ a 2-ring, def. \ref{2RingAsCompatiblyMonoidalPresentableCategory}, write

\begin{displaymath}
2Mod_A \in 2Cat
\end{displaymath}
for the [[2-category]] of [[module objects]] over $A$ in $2Ab$.

This means that a 2-module over $A$ is a [[presentable category]] $N$ equipped with a functor

\begin{displaymath}
A \boxtimes N \to N
\end{displaymath}
which satisfies the evident action property.

\end{defn}
(\hyperlink{CJF}{CJF, def. 2.3.3})

\begin{example}
\label{}\hypertarget{}{}
Let $R$ be an ordinary [[commutative ring]] and $A$ an ordinary $R$-[[associative algebra|algebra]]. Then by example \ref{CategoryOfModulesAs2AbelianGroup} $Mod_A$ is a 2-abelian group and by example \ref{CommutativeRingGivesCommutative2Ring} $Mod_R$ is a commutative ring. By example \ref{TensorProductModuleCategoriesAsOf2AbelianGroups} $Mod_R$-[[2-module]] structures on $Mod_A$

\begin{displaymath}
Mod_R \boxtimes \Mod_A \to Mod_A
\end{displaymath}
correspond to colimit-preserving functors

\begin{displaymath}
Mod_{R \otimes_{\mathbb{Z}} A} \to Mod_{A}
\end{displaymath}
that satisfy the action property. Such as presented under the [[Eilenberg-Watts theorem]], prop. \ref{EilenbergWattsTheorem}, by $R \otimes_{\mathbb{Z}} A$-$A$ [[bimodules]]. $A$ itself is canonically such a bimodule and it exhibits a $Mod_R$-[[2-module]] structure on $Mod_A$.

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

\hypertarget{PropertiesInitialObject}{}\subsubsection*{{Initial object}}\label{PropertiesInitialObject}

\begin{remark}
\label{}\hypertarget{}{}
The analog role in 2-rigs to the role played by the [[natural numbers]] among ordinary [[rigs]] should be played by the standard [[categorification]] of the natural numbers: the [[category of finite sets]]. One is therefore inclined to demand that a reasonable definition of 2-rigs should be such that $FinSet$ is the [[initial object]] (in the suitably higher categorical sense) in the [[2-category]] of 2-rigs.

For the notion in def. \ref{BD2Rig} this was conjectured by [[John Baez]], for the notion in def. \ref{2RingAsCompatiblyMonoidalPresentableCategory} this is asserted in (\hyperlink{CJF}{Chirvasitu \& Johnson-Freyd, example 2.3.4}).

\end{remark}
\hypertarget{TannakaDuality}{}\subsubsection*{{Tannaka duality}}\label{TannakaDuality}

[[!include structure on algebras and their module categories - table]]

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

\begin{itemize}%
\item [[2-algebraic geometry]]


\item A further slight variant of compatibly monoidal cocomplete categories is that of \emph{monoidal [[vectoids]]}.


\item [[distributivity for monoidal structures]]


\item [[prime spectrum of a monoidal stable (∞,1)-category]]



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

The proposal that a 2-ring should be a compatibly monoidal cocomplete category is due to

\begin{itemize}%
\item [[John Baez]], [[James Dolan]], \emph{Higher-dimensional algebra III: $n$-categories and the algebra of opetopes}, \emph{Adv. Math.} \textbf{135} (1998), 145-206. (\href{http://arxiv.org/abs/q-alg/9702014}{arXiv})

\end{itemize}
The proposal that a 2-ring should be a compatibly monoidal presentable category is due to

\begin{itemize}%
\item [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], \emph{The fundamental pro-groupoid of an affine 2-scheme} (\href{http://arxiv.org/abs/1105.3104}{arXiv:1105.3104})

\end{itemize}
see also

\begin{itemize}%
\item [[Martin Brandenburg]], \emph{Tensor categorical foundations of algebraic geometry} (\href{http://arxiv.org/abs/1410.1716}{arXiv:1410.1716})

\end{itemize}
This is related to

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Tannaka duality for geometric stacks]]}.

\end{itemize}
A similar notion is that of ``monoidal [[vectoid]]'' due to

\begin{itemize}%
\item [[Nikolai Durov]], \emph{Classifying vectoids and generalisations of operads}, Proc. of Steklov Inst. of Math. \textbf{273}:1, 48-63 (2011) \href{http://arxiv.org/abs/1105.3114}{arxiv/1105.3114}), the translation of ``    '', Trudy MIAN, vol. 273

\end{itemize}
The role of presentable categories as higher analogs abelian groups in the context of [[(infinity,1)-categories]] have been made by [[Jacob Lurie]], see at \emph{[[Pr(infinity,1)Cat]]}.

Another, more algebraic, notion of a categorical ring is introduced in

\begin{itemize}%
\item M. Jibladze , T. Pirashvili, \emph{Third Mac Lane cohomology via categorical rings}, J. of homotopy and related structures, \textbf{2}(2), 2007, 187--221 \href{http://www.emis.de/journals/JHRS/volumes/2007/n2a10/v2n2a10.pdf}{pdf} \href{http://arxiv.org/abs/math/0608519}{math.KT/0608519}

\end{itemize}
[[!redirects 2-rig]] [[!redirects 2-rigs]] [[!redirects 2-ring]] [[!redirects 2-rings]]

[[!redirects 2Rig]] [[!redirects 2Ring]]



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