nLab subtractive category



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

Here a relation r:RX×Yr\colon R\to X\times Y is left (respectively right) punctual if (id X,0):XX×Y(id_X,0)\colon X\to X\times Y (respectively (0,id Y):YX×Y(0,id_Y)\colon Y\to X\times Y) factors through rr.


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 ss and constant 00 such that s(x,x)=0s(x,x)=0 and s(x,0)=xs(x,0)=x.

Relation to Malcev categories

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

Here Pt(B)Pt(B) is the pointed category whose objects are triples (AC,α:AB,β:BA)(A\in C,\alpha\colon A\to B,\beta\colon B\to A) such that αβ=id B\alpha\circ\beta=id_B. Morphisms (A,α,β)(A,α,β)(A,\alpha,\beta)\to(A',\alpha',\beta') are morphisms f:AAf\colon A\to A' such that αf=α\alpha'\circ f=\alpha and fβ=βf\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.

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