What is a theory of exactness properties?

It is a general study of exactness properties along the following lines:

• Identify a class of exactness properties.
• Establish theorems that are valid for arbitrary exactness properties in the class.
• Establish structure theorems: how do these properties, or categories satisfying them, interact with each other?
  

Regular matrix properties

• These are properties of regular categories given by sequents in regular logic involving a single predicate (ZJ - 2008). Examples of such properties are the defining properties of: Mal’tsev, majority, arithmetical and n-permutable regular categories. In the context of pointed categories, further examples include defining properties of: unital, subtractive, and strongly unital regular categories. These properties correspond to linear Mal’tsev conditions in universal algebra (ZJ - 2008).
• Unified theorems for such properties include characterisation of essentially algebraic categories having these properties and embedding theorems, making use of a power of an essentially algebraic category determined by the property (PAJ 2016, 2020).
• Another interesting result is that these properties are stable under exact completion of a regular category (PAJ 2020).
• These properties turn out to be precisely those exactness properties which state that a canonical morphism between finite limits built from a finite diagram is a regular epimorphism (PAJ & ZJ - 2021).
  

Jonsson terms (1967)

Example:

Congruence

n-distributivity

Lex-cocompleteness exactness properties

• These are properties of categories given by compatibility of a particular type of colimits with finite limits, as fulfilled in the category of sets (RG & SL - 2012). Examples of such properties include defining properties of: regular, Barr-exact, lextensive (finitely complete extensive), coherent, geometric and adhesive categories.
• Unified theorems for such properties include completion and embedding theorems: any finitely complete category can be completed to one having necessary colimits that behave as the property requires (RG & SL - 2012); a finitely complete category has the property if and only if it admits a full embedding in a Grothendieck topos that preserves the colimits specified in the property.
• Implication of such properties gives rise to an adjunction between the 2-categories of categories having the properties (RG & SL - 2012).
  

Example:

Finitely complete extensive categories

Carboni - Lack - Walters 1993

Pro-exactness properties

• These are properties of categories which formulate a certain behaviour of finite (finite) limits and colimits constructed from a given diagram (PAJ & ZJ - 2021). Examples of such properties include all matrix properties under regularity, as well as the defining properties of: pointed, linear, regular, Barr-exact Mal’tsev, regular protomodular with binary coproducts, regular involution-rigid with binary coproducts, weakly May’tsev with binary coproducts, semi-abelian, abelian, additive, normal, semi-abelian with Smith is Huq condition as well as with normality of Higgins commutator condition as well as with algebraic coherence, coherent with finite coproducts, distributive, extensive with pullbacks categories.
• Unified theorems for such properties include an embedding theorem, making use of the cofiltered limit completion of a finitely complete category (PAJ & ZJ - 2021).
• These properties are defined using the language of sketches.
  

Unconditional representation of pullback stability of regular epimorphisms

Example:

Regularity

Algebraic exactness properties

• These are properties of categories which, roughly speaking, state that in a diagram where objects are built using (finite) limits and colimits, a morphism to a limit of primitive objects is a strong epimorphism (PAJ - 2022). Examples of such properties include all matrix properties, as well as the defining properties of: protomodular, involution-rigid, normal, and normal-projection categories.
• Unified theorems for such properties include characterisation of finitely bicomplete categories using approximation morphisms (PAJ - 2022).
• Conjecture: these properties are all and only categorical extensions of Mal’tsev conditions. Problem: describe Mal’tsev conditions captured this way.
  

Example:

Protomodularity via approximation morphisms

Bourn - 1991

Bourn - G. Janelidze 2003

Bourn - Z. Janelidze 2011

Jacqmin 2022

Matrix properties

• These are predecessors of regular matrix properties, formulated for finitely complete categories (ZJ - 2004, 2006) using special types of sequents in cartesian logic. Examples of such properties are the defining properties of: Mal’tsev, majority, and arithmetical categories.
• Unified theorems for such properties include Bourn localisation theorems (ZJ - 2006), as well as an algorithm for deciding their implication (PAC & ZJ & MH - 2022).
• For binary matrices, the arithmetical property is the smallest property, while the upset of the majority property and the down-set of the Mal’tsev property (each of which are infinite) cover all others. Note also that the arithmetical property is the intersection of the Mal’tsev property and the majority property (PAC & ZJ & MH & EvdW - 2023).
• For diagonal matrices, the poset of diagonal matrix properties has been fully computed (PAC & ZJ & MH & EvdW - 2023).
  

Example:

Deriving majority from arithmeticity using the algorithm

Geometric derivation

Algebraic derivation

The poset of diagonal matrix properties

Determining the structure of the (infinite) poset of all binary matrix properties is an open problem!

First-order exactness properties

• These are properties that can be encoded using a single surjective sketch morphism (PAJ & ZJ - 2021). They include all exactness properties mentioned thus far (that lex-cocompleteness exactness properties are such is conjectured).
• Bicontinuous conservative faithful functors reflect such properties.
• Open question: is it possible to describe varieties of bicomplete categories having such properties? (Similar questions can be formulated for interesting classes of exactness properties)
• Open question: which Mal’tsev conditions are equivalent to such exactness properties?
  

Primitive quasi-equational exactness properties

• This is a new type of exactness properties that are given by quasi-identities of equations involving a single functional symbol (ZJ & MH).
• Every such exactness property is algebraic and a Mal’tsev condition.
  
• Open question: is there an algorithm for deciding implications of these properties?
  

Examples:

• f(x,0) = 0 ==> f(x,x) = f(0,x) — categories with normal projections (AxB/A = B): unital categories, subtractive categories, dual of the category of monoids where the identity element has no proper divisors (xy = 1 implies x = 1 and y = 1), algebras with p(x,0) = x and p(0,x) =
• p(x,x)
• f(x,0) = 0 = f(0,x) ==> f(x,x) = 0 — “nullital” categories (A/(BxC) = (A/B)/C): unital categories, pointed varieties having difunctional class relations, algebras with p(x,0) = p(0,x) and p(x,0) = 0 => x = 0.