\[ \require{mathtools} \let\DeclarePairedDelimiter\DeclarePairedDelimiters % MathJax typo %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % symbol %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % sets \newcommand{\N}{\mathbb{N}} % natural numbers \newcommand{\Z}{\mathbb{Z}} % integers \newcommand{\Q}{\mathbb{Q}} % rational numbers \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\C}{\mathbb{C}} % complex numbers % category \newcommand{\catf}[1]{{\mathbf{#1}}} \newcommand{\cA}{\catf{A}} \newcommand{\cB}{\catf{B}} \newcommand{\cC}{\catf{C}} \newcommand{\cD}{\catf{D}} \newcommand{\cE}{\catf{E}} \newcommand{\cF}{\catf{F}} \newcommand{\cG}{\catf{G}} \newcommand{\cH}{\catf{H}} \newcommand{\cI}{\catf{I}} \newcommand{\cJ}{\catf{J}} \newcommand{\cK}{\catf{K}} \newcommand{\cL}{\catf{L}} \newcommand{\cM}{\catf{M}} \newcommand{\cN}{\catf{N}} \newcommand{\cO}{\catf{O}} \newcommand{\cP}{\catf{P}} \newcommand{\cQ}{\catf{Q}} \newcommand{\cR}{\catf{R}} \newcommand{\cS}{\catf{S}} \newcommand{\cT}{\catf{T}} \newcommand{\cU}{\catf{U}} \newcommand{\cV}{\catf{V}} \newcommand{\cW}{\catf{W}} \newcommand{\cX}{\catf{X}} \newcommand{\cY}{\catf{Y}} \newcommand{\cZ}{\catf{Z}} \newcommand{\cZero}{\catf{0}} \newcommand{\cOne}{\catf{1}} \newcommand{\cTwo}{\catf{2}} \newcommand{\cArr}{\catf{Arr}} % arrow \newcommand{\cPSh}{\catf{PSh}} % presheaf \newcommand{\cCone}{\catf{Cone}} % cone \newcommand{\cCocone}{\catf{Cocone}} % cocone \newcommand{\cFin}{\catf{Fin}} % finite prefix \newcommand{\cSet}{\catf{Set}} % functions \newcommand{\cRel}{\catf{Rel}} % relations \newcommand{\cPos}{\catf{Pos}} % posets \newcommand{\cMon}{\catf{Mon}} % monoids \newcommand{\cCMon}{\catf{CMon}} % commutative monoids \newcommand{\cGrp}{\catf{Grp}} % groups \newcommand{\cAb}{\catf{Ab}} % abelian groups \newcommand{\cCat}{\catf{Cat}} % categories \newcommand{\cGrpd}{\catf{Grpd}} % groupoids \newcommand{\cVect}{\catf{Vect}} % vector spaces \newcommand{\cMat}{\catf{Mat}} % matrices \newcommand{\cTop}{\catf{Top}} % topological spaces and continuous maps \newcommand{\cMet}{\catf{Met}} % metric spaces and metric maps \newcommand{\cMeas}{\catf{Meas}} % measurable spaces and measurable functions \newcommand{\cStoch}{\catf{Stoch}} % measurable spaces and stochastic maps \newcommand{\cProb}{\catf{Prob}} % probability measures and measure-preserving functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % supscript %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\op}{^\mathrm{op}} \newcommand{\inv}{^{-1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % function %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\logsumexp}{log-sum-exp} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % arrow %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % morphism \newcommand{\xto}{\xrightarrow} % -> \newcommand{\xot}{\xleftarrow} % <- \newcommand{\toot}{\rightleftarrows} % <=> \newcommand{\xtoot}[2]{\xrightleftharpoons[#2]{#1}} \newcommand{\iso}{\cong} % ~= \newcommand{\klto}{\rightsquigarrow} % ~> \newcommand{\mono}{\rightarrowtail} % >-> \newcommand{\epi}{\twoheadrightarrow} % ->> \newcommand{\ihom}{\multimap} % -o % category \newcommand{\incl}{\hookrightarrow} \newcommand{\adjto}[2]{\overset{{}\xto[]{#1}{}}{\underset{{}\xot[#2]{}{}}{\bot}}} % functor \newcommand{\nat}{\Rightarrow} % => \newcommand{\xnat}{\xRightarrow} % => \newcommand{\comma}{\downarrow} \newcommand{\adj}{\dashv} % -| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % delimiter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclarePairedDelimiter{\parens}{(}{)} % parentheses ( ) \DeclarePairedDelimiter{\bracks}{[}{]} % brackets [ ] \DeclarePairedDelimiter{\braces}{\{}{\}} % braces { } \DeclarePairedDelimiter{\angles}{\langle}{\rangle} % angles \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} \newcommand{\set}{\braces} \newcommand{\singleton}{{\set{*}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % operator %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\compL}{\mathbin{\circ}} \newcommand{\compR}{\mathbin{;}} \newcommand{\Kl}{\mathrm{Kl}} \DeclareMathOperator{\Obj}{Obj} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sub}{Sub} \DeclareMathOperator{\Eq}{Eq} \DeclareMathOperator{\Coeq}{Coeq} \DeclareMathOperator{\Pull}{Pull} \DeclareMathOperator{\Push}{Push} \DeclareMathOperator{\Lan}{Lan} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\Lift}{Lift} \DeclareMathOperator{\Rift}{Rift} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\act}{act} \DeclareMathOperator{\colim}{colim} % exponential \DeclareMathOperator{\partf}{partial} \DeclareMathOperator{\curry}{curry} % natural number \DeclareMathOperator{\zero}{zero} \DeclareMathOperator{\successor}{succ} % list \DeclareMathOperator{\nil}{nil} \DeclareMathOperator{\cons}{cons} \newcommand{\concat}{\mathbin{{+}\mspace{-8mu}{+}}} % fold \DeclareMathOperator{\fold}{fold} \DeclareMathOperator{\map}{map} \DeclareMathOperator{\filter}{filter} % bool \DeclareMathOperator{\cond}{cond} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % order %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\meet}{\wedge} \newcommand{\join}{\vee} \newcommand{\bigmeet}{\bigwedge} \newcommand{\bigjoin}{\bigvee} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % logic %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\lfal}{\bot} \newcommand{\ltru}{\top} \newcommand{\lcon}{\wedge} \newcommand{\lncon}{\uparrow} \newcommand{\ldis}{\vee} \newcommand{\lndis}{\downarrow} \newcommand{\limp}{\rightarrow} \newcommand{\lnimp}{\nrightarrow} \newcommand{\lcim}{\leftarrow} \newcommand{\lncim}{\nleftarrow} \newcommand{\leqv}{\leftrightarrow} \newcommand{\lneqv}{\nleftrightarrow} \newcommand{\lproves}{\vdash} % |- \newcommand{\lmodels}{\vDash} % |= \newcommand{\lnproves}{\nvdash} % |/- \newcommand{\lnmodels}{\nvDash} % |/= \]

1 Enriched category

1.1 \(\cSet_*\)-enriched category

1.2 \(\cCMon\)-enriched category

1.3 \(\cAb\)-enriched category

2 Matrix category

2.1 Semiring, ring, and field

A semiring (rig) \((R, +, \times, 0, 1)\) is a set \(R\) equipped with two binary operations, addition \(+\) and multiplication \(\times\), and two distinguished elements \(0\) and \(1\), such that

• addition \((R, +, 0)\) is a commutative monoid,
• multiplication \((R, \times, 1)\) is a monoid,
• multiplication \(\times\) distributes over addition \(+\), and
• the additive identity \(0\) annihilates multiplication \(\times\): \(0 \times a = 0 = a \times 0\).

An additively idempotent semiring is a semiring \((R, +, \times, 0, 1)\) such that addition \(+\) is idempotent: \(a + a = a\).

\(1 + 1 = 1\) implies \(a + a = a \times (1 + 1) = a \times 1 = a\).

A multiplicatively idempotent semiring is a semiring \((R, +, \times, 0, 1)\) such that multiplication \(\times\) is idempotent: \(a \times a = a\).

• natural number semiring \((\N, +, \times, 0, 1)\)
• Boolean semiring \((\set{\lfal, \ltru}, \ldis, \lcon, \lfal, \ltru) \iso (\set{0, 1}, \max, \min, 0, 1) \incl ([0, 1], \max, \min, 0, 1)\) max-min semiring
• min-plus semiring \(([0, \infty], \min, +, \infty, 0) \xtoot{\exp(-x)}{-\log(x)} ([0, 1], \max, \times, 0, 1)\) Viterbi semiring
• min-plus semiring \(((-\infty, \infty], \min, +, \infty, 0) \xtoot{-}{-} ([-\infty, \infty), \max, +, -\infty, 0)\) max-plus semiring
• probability semiring \(([0, \infty), +, \times, 0, 1) \xtoot{\log}{\exp} ([-\infty, \infty), \logsumexp, +, -\infty, 0)\) log semiring
  • \(\log(a + b) = \logsumexp(\log a, \log b)\)
  • \(\log(a \times b) = \log a + \log b\)
  • \(\log 0 = -\infty\)
  • \(\log 1 = 0\)
  • \(\exp(\logsumexp(a, b)) = \exp a + \exp b\)
  • \(\exp(a + b) = \exp a \times \exp b\)
  • \(\exp(-\infty) = 0\)
  • \(\exp 0 = 1\)

A ring \((R, +, \times, 0, 1)\) is a set \(R\) equipped with two binary operations, addition \(+\) and multiplication \(\times\), and two distinguished elements \(0\) and \(1\), such that

• addition \((R, +, 0)\) is an abelian group,
• multiplication \((R, \times, 1)\) is a monoid, and
• multiplication \(\times\) distributes over addition \(+\).

In a ring, the annihilation law follows from the distributivity law: \(0 \times a + 0 \times a = (0 + 0) \times a = 0 \times a\), so \(0 \times a = 0\).

A Boolean ring is a multiplicatively idempotent ring.

• integer ring \((\Z, +, \times, 0, 1)\)
• Boolean ring \((\set{\lfal, \ltru}, \lneqv, \ldis, \lfal, \ltru) \iso (\Z/2\Z, +, \times, 0, 1)\)

A field \((R, +, \times, 0, 1)\) is a set \(R\) equipped with two binary operations, addition \(+\) and multiplication \(\times\), and two distinguished elements \(0\) and \(1\), such that

• addition \((R, +, 0)\) is an abelian group,
• multiplication \((R, \times, 1)\) is an abelian group, and
• multiplication \(\times\) distributes over addition \(+\).

• rational number field \((\Q, +, \times, 0, 1)\)
• real number field \((\R, +, \times, 0, 1)\)
• complex number field \((\C, +, \times, 0, 1)\)

2.2 Matrix operations

For natural numbers \(a, b \in \N\) and a set \(X\), a \(X\)-valued matrix \(M_{a \times b}\) of dimension \(a \times b\) is a function \(M: [a] \times [b] \to X\), where \([a] := \set{i \in \N \mid 1 \leq i \leq a}\). \(M(i, j)\) is also denoted by \(M_{ij}\).

The empty matrix \(\varnothing_{0 \times a}\) or \(\varnothing_{a \times 0}\) is the unique function from the empty set \(\varnothing\) to \(X\).

The semiring \(R\)-valued identity matrix \(I_a\) is an \(a \times a\) matrix such that \(I_a(i, j) = 1\) if \(i = j\) and \(I_a(i, j) = 0\) if \(i \neq j\).

The semiring \(R\)-valued zero matrix \(0_{a \times b}\) is an \(a \times b\) matrix such that \(0_{a \times b}(i, j) = 0\).

The matrix product of two semiring \(R\)-valued matrices \(M_{a \times b}\) and \(N_{b \times c}\) is a \(R\)-valued matrix \((MN)_{a \times c}\) such that

\[(MN)_{ik} := \sum_j M_{ij} \times N_{jk}.\]

Specifically, \(\varnothing_{a \times 0} \varnothing_{0 \times c} = 0_{a \times c}\).

The matrix addition of two semiring \(R\)-valued matrices \(M_{a \times b}\) and \(N_{a \times b}\) is a \(R\)-valued matrix \((M + N)_{a \times b}\) such that

\[(M + N)_{ij} := M_{ij} + N_{ij}.\]

The left scalar multiplication of a semiring \(R\)-valued matrix \(M_{a \times b}\) and a scalar \(r \in R\) is a \(R\)-valued matrix \((rM)_{a \times b}\) such that

\[(rM)_{ij} := r \times M_{ij}.\]

The Kronecker product of two semiring \(R\)-valued matrices \(M_{a \times b}\) and \(N_{c \times d}\) is a \(R\)-valued matrix \((M \otimes N)_{(a \times c) \times (b \times d)}\) such that

\[\begin{aligned} (M \otimes N) := \begin{bmatrix} M_{11} N & \cdots & M_{1b} N \\ \vdots & \ddots & \vdots \\ M_{a1} N & \cdots & M_{ab} N \end{bmatrix}_{(a \times c) \times (b \times d)}. \end{aligned}\]

2.3 Matrix category

The matrix category \(\cMat(R)\) over a semiring \(R\):

• objects: \(\N\)
• morphisms from \(a\) to \(b\): \(a \times b\) \(R\)-valued matrices
• composition: matrix multiplication
• identity: identity matrix \(I_a\)

2.3.1 Zero object

• zero object: \(0\)
• terminal morphism: \(\varnothing_{a \times 0}\)
• initial morphism: \(\varnothing_{0 \times a}\)
• zero morphism: zero matrix \(0_{a \times b}: a \xto{\varnothing_{a \times 0}} 0 \xto{\varnothing_{0 \times b}} b\)

2.3.2 Biproduct: addition

• \(a \oplus b := a + b\)
• \(\angles{M, N} := \begin{bmatrix} M & N\end{bmatrix}_{c \times (a + b)}\)
• \(\bracks{M, N} := \begin{bmatrix} M \\ N\end{bmatrix}_{(a + b) \times c}\)
• \(p_1 = \begin{bmatrix} I_a \\ 0_{b \times a} \end{bmatrix}_{(a + b) \times a}\)
• \(p_2 = \begin{bmatrix} 0_{a \times b} \\ I_b \end{bmatrix}_{(a + b) \times b}\)
• \(i_1 = \begin{bmatrix} I_a & 0_{a \times b} \end{bmatrix}_{a \times (a + b)}\)
• \(i_2 = \begin{bmatrix} 0_{b \times a} & I_b \end{bmatrix}_{b \times (a + b)}\)

• \(i_1 p_1 = I_a\)
• \(i_2 p_2 = I_b\)
• \(i_1 p_2 = 0_{a \times b}\)
• \(i_2 p_1 = 0_{b \times a}\)
• \(p_1 i_1 + p_2 i_2 = \begin{bmatrix} I_a & 0_{a \times b} \\ 0_{b \times a} & 0_{b \times b} \end{bmatrix}_{(a + b) \times (a + b)} + \begin{bmatrix} 0_{a \times a} & 0_{a \times b} \\ 0_{b \times a} & I_b \end{bmatrix}_{(a + b) \times (a + b)} = I_{a + b}\)

2.3.3 Biproduct morphism: block diagonal matrix

\[\begin{aligned} M_{a \times b} \oplus N_{c \times d} = \begin{bmatrix} M_{a \times b} & 0_{a \times d} \\ 0_{c \times b} & N_{c \times d} \end{bmatrix}_{(a + c) \times (b + d)} \end{aligned}\]

2.3.4 Diagonal & codiagonal: concatenation

• \(\Delta_a = \begin{bmatrix} I_a & I_a \end{bmatrix}_{a \times (a + a)}\)
• \(\nabla_a = \begin{bmatrix} I_a \\ I_a \end{bmatrix}_{(a + a) \times a}\)

2.3.5 Addition: elementwise sum

For \(M_1, M_2 \in \Hom(a, b)\), \(M_1 + M_2 = \Delta_a (M_1 \oplus M_2) \nabla_b\).

• \((A + B)C = AC + BC\)
• \(A(B + C) = AB + AC\)

2.3.6 Monoidal product: multiplication & Kronecker product

• multiplication \(a \otimes b := a \times b\)
• Kronecker product \(M \otimes N\)
• unit \(1\)
• composition: \((MN) \otimes (M'N') = (M \otimes M')(N \otimes N')\)
• identity: \(I_a \otimes I_b = I_{a \times b}\)
• associativity: \(a \times (b \times c) = (a \times b) \times c\), \(A \otimes (B \otimes C) = (A \otimes B) \otimes C\)
• unitality: \(1 \times a = a = a \times 1\), \(I_1 \otimes A = A = A \otimes I_1\)
• left distributivity: \(a \times (b + c) = a \times b + a \times c\), \((A \oplus B) \otimes C = A \otimes C \oplus B \otimes C\)
• right distributivity: \((a + b) \times c = a \times c + b \times c\), \(A \otimes (B \oplus C) = A \otimes B \oplus A \otimes C\)

3 Preadditive category

4 Additive category

5 Preabelian category

6 Abelian category

• 1 Enriched category
  • 1.1 \(\cSet_*\)-enriched category
  • 1.2 \(\cCMon\)-enriched category
  • 1.3 \(\cAb\)-enriched category
• 2 Matrix category
  • 2.1 Semiring, ring, and field
  • 2.2 Matrix operations
  • 2.3 Matrix category
• 3 Preadditive category
• 4 Additive category
• 5 Preabelian category
• 6 Abelian category