category theory

# Contents

## Definition

A category is subtractive if it is pointed, admits finite limits, and left punctual reflexive relations are right punctual.

Here a relation $r\colon R\to X\times Y$ is left (respectively right) punctual if $(id_X,0)\colon X\to X\times Y$ (respectively $(0,id_Y)\colon Y\to X\times Y$) factors through $r$.

## Example

A variety of algebras is a subtractive category if and only it is a subtractive variety in the sense of Ursini, meaning its theory contains a binary operation $s$ and constant $0$ such that $s(x,x)=0$ and $s(x,0)=x$.

## Relation to Malcev categories

A category $C$ with finite limits is a Malcev category if and only if for any object $B$ of $C$ the category $Pt(B)$ is subtractive.

Here $Pt(B)$ is the pointed category whose objects are triples $(A\in C,\alpha\colon A\to B,\beta\colon B\to A)$ such that $\alpha\circ\beta=id_B$. Morphisms $(A,\alpha,\beta)\to(A',\alpha',\beta')$ are morphisms $f\colon A\to A'$ such that $\alpha'\circ f=\alpha$ and $f\circ\beta=\beta'$.

• Zurab Janelidze, Subtractive categories, Appl. Categ. Struct 13, No. 4, 343-350 (2005) (doi:10.1007/s10485-005-0934-8)

• Zurab Janelidze, Closedness properties of internal relations. A series of papers.

• I. A unified approach to Malʹtsev, unital and subtractive categories.

• II. Bourn localization.

• III. Pointed protomodular categories.

• IV. Expressing additivity of a category via subtractivity.

• V. Linear Malʹtsev conditions.

• VI. Approximate operations.