Exact sequences and irreversibility

Recently, I defined morphism sequences and related concepts in any abstract category. There are many more concepts that can be defined categories with additional structure such as abelian categories, and exact sequences are a particularly important example. The key to the construction is that abelian categories have certain limit/colimit related properties like the existence of kernels and cokernels for any morphism.

Ordering relations and their subrelations are characterized by the fact that given any two distinct comparable elements, there is no reverse path from the sucessor element to its predecessor. In other words, partial orders are directed acyclic graphs and acyclicity is path-irreversibility. In the other direction, morphisms are associated with degrees of irreversibility. This leads to the dual logics of set theory and partition logic in concrete categories.

Take the category of sets as an example: an irreversible morphism $f: A \to B$ uses a subset of the input information of $A$ and a subset of the output values of $B$. That is, whenever we are using an irreversible morphism, you are implicitly referring to an order relation. In the other direction, category theorists like to relate order to irreversible morphisms, in particular through monomorphisms and epimorphisms. Proper subobjects are simply defined by irreversible mononomorphisms. All category-theoretic concepts of order simply come from different notions of irreversibility.

I would characterize exact sequences as a construction of this sort. The essence of exact sequences lies in how they allow for certain manipulations with regards to degrees of reversibility. In doing so, they allow for a convenient expression of order-related concepts like group extensions in certain categories. In particular, while in the category of sets there are two different kinds of reversibility covered by partition logic and set theory the concept of a kernel in an abelian category allows a relation between the two types of irreversibility.

Exact pairs:
Binary relations of morphisms are one of the foundational concepts of category theory, in particular composition is defined on the relation $M^2_{*}$ of all inner-equal ordered pairs of morphisms because composition is a partial operation. In that same vein, I would define exactness as a binary relation between morphisms: two inner-equal morphisms in an abelian category are exact if the image of the first morphism is equal to the kernel of the next morphism.

To see how this exactness construction is simply a trick for manipulating degrees of reversibility, consider that $1 \to A \to B$ simply makes the morphism from $A$ to $B$ a monomorphism (and therefore an expression of subobjects) and $A \to B \to 1$ makes it into an epimorphism (and therefore an expression of quotients). Therefore, simple exact sequences allow you to express the dual order-theoretic irreversibility-related concepts of ordering in a category using morphisms.

By combining both constructions we can clearly make a morphism reversible. For example, the exact sequence $1 \to A \to B \to 1$ has a reversible central morphism. Reversibility is ensured because in an abelian category, every bimorphism is an isomorphism. In this case, no proper order-theoretic concept like a subobject or a quotient is described but rather what is described is the lack of category-theoretic distinctions.

Exact sequences and group extensions:
Category-theoretic irreversibility allows for previously order-theoretic concepts like subobjects and quotients to be described in category-theoretic language. Using exact sequences we can describe properties of both subobjects and quotients at once. The decisive example comes from group extensions, as exact sequences can be used to describe how a quotient group emerges from a normal subgroup of a group. \[ 1 \to N \to G \to Q \to 1 \] This is the most common construction, and it provides a different language for expressing the already widely familiar concept of a group extension. The reason that this is possible is because of how exactness relates to the irreversibility orderings of a category. These irreversibility orderings lead to the dual notions of subobjects and quotients.