Normalization equivalence, kernel equivalence and affine categories. (English) Zbl 0756.18007

Bourn, Dominique

Category theory, Proc. Int. Conf., Como/Italy 1990, Lect. Notes Math. 1488, 43-62 (1991).

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[For the entire collection see Zbl 0733.00009].
It is shown here, that for a left exact category \(\mathbb{E}\) with an initial object and O-valued sums, kernel equivalence is equivalent to the following condition (called the essentially affine condition): for any commutative square of split epimorphisms: \[ \begin{tikzcd}\cdot\ar[r] & \cdot \\ \cdot \ar[r]\ar[u,"\Downarrow"] & \cdot \ar[u,"\Downarrow"']\end{tikzcd} \] the downward square is a pullback if and only if the upward square is a pushout. Now a left exact category is additive if and only if it essentially affine and pointed \((0=1)\). It is modular if and only if it is essentially affine and its terminal object satisfies a condition of modularity. It is equivalent to a coslice of additive category if and only if it is essentially affine and its initial object satisfies a certain condition of comodularity.

MSC:

18E10	Abelian categories, Grothendieck categories
18E05	Preadditive, additive categories
18A30	Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)

Keywords:

categories of affine spaces; slices of additive categories; left exact category; kernel equivalence; essentially affine condition; split epimorphisms; pullback; pushout; modularity; coslice; additive category; comodularity

Citations:

Zbl 0683.18008; Zbl 0733.00009