On the decomposition of a strong epimorphism into regular epimorphisms

The fundamental theorem of homomorphisms plays a central role in abstract algebra. It states that for every homomorphism, the quotient algebra modulo its kernel is isomorphic to its image, a subalgebra of its codomain. In category theory, the theorem can be reformulated as a decomposition of a morphism into a regular epimorphism (= quotient) followed by a monomorphism (= subalgebra). However, as the notion of algebras is generalized beyond classical equational ones, such a decomposition may not be possible. Indeed, the decomposition tends to fail in several categories of partial algebras, algebras including partially defined operations.

The category $\mathrm{𝐂𝐚𝐭}$ of (small) categories is one of the typical examples of the category of partial algebras. Consider the following morphism (= functor) in $\mathrm{𝐂𝐚𝐭}$:

Then, the kernel of the functor $\mathrm{\Phi }$ identifies the objects $B$ and $B^{\prime }$ in ${𝒞}_1$, but there is no way to identify $g\circ f$ and $h$ inside the domain category ${𝒞}_1$ because $g\circ f$ does not even make sense there. In short, the morphism collapses beyond what the domain category can see. Once $B$ and $B^{\prime }$ get identified (in ${𝒞}_2$), $g\circ f$ comes to make sense, and taking another quotient by the equality $g\circ f=h$ gives rise to its image.

The decomposition number $\delta (f)$ for a morphism $f$, introduced in [1], is the number of iterations of taking quotients needed to obtain an object isomorphic to the image. For example, we have $\delta (\mathrm{\Phi })=2$ in the above example. In fact, for an arbitrary functor $\mathrm{\Psi }$, we have $\delta (\mathrm{\Psi })\le 2$. Introducing a syntactic way to give an upper bound for the decomposition numbers, we will demonstrate several examples of calculating the supremum length of such decompositions and will present further generalizations. This talk is based on a paper [2] in preparation.