Pseudoelements in abelian categories

A pseudoelement of an object X in an abelian category C is an equivalence class of arrows ending in X, where two arrows are considered equivalent if we can find two epimorphisms with a common domain making a commutative square with the two arrows. While the construction shows that pseudoelements are actually subobjects of X rather than "elements", it is possible to chase these pseudoelements through commutative diagrams in an abelian category to prove exactness properties. This is done using some "diagram-chasing metatheorems" proved in this file. In many cases, a proof in the category of abelian groups can more or less directly be converted into a proof using pseudoelements.

A classic application of pseudoelements are diagram lemmas like the four lemma or the snake lemma.

Pseudoelements are in some ways weaker than actual elements in a concrete category. The most important limitation is that there is no extensionality principle: If f g : X ⟶ Y, then ∀ x ∈ X, f x = g x does not necessarily imply that f = g (however, if f = 0 or g = 0, it does). A corollary of this is that we can not define arrows in abelian categories by dictating their action on pseudoelements. Thus, a usual style of proofs in abelian categories is this: First, we construct some morphism using universal properties, and then we use diagram chasing of pseudoelements to verify that is has some desirable property such as exactness.

It should be noted that the Freyd-Mitchell embedding theorem gives a vastly stronger notion of pseudoelement (in particular one that gives extensionality). However, this theorem is quite difficult to prove and probably out of reach for a formal proof for the time being.

Main results

We define the type of pseudoelements of an object and, in particular, the zero pseudoelement.

We prove that every morphism maps the zero pseudoelement to the zero pseudoelement (apply_zero) and that a zero morphism maps every pseudoelement to the zero pseudoelement (zero_apply)

Here are the metatheorems we provide:

• A morphism f is zero if and only if it is the zero function on pseudoelements.
• A morphism f is an epimorphism if and only if it is surjective on pseudoelements.
• A morphism f is a monomorphism if and only if it is injective on pseudoelements if and only if ∀ a, f a = 0 → f = 0.
• A sequence f, g of morphisms is exact if and only if ∀ a, g (f a) = 0 and ∀ b, g b = 0 → ∃ a, f a = b.
• If f is a morphism and a, a' are such that f a = f a', then there is some pseudoelement a'' such that f a'' = 0 and for every g we have g a' = 0 → g a = g a''. We can think of a'' as a - a', but don't get too carried away by that: pseudoelements of an object do not form an abelian group.

Notations

We introduce coercions from an object of an abelian category to the set of its pseudoelements and from a morphism to the function it induces on pseudoelements.

These coercions must be explicitly enabled via local instances: local attribute [instance] object_to_sort hom_to_fun

Implementation notes

It appears that sometimes the coercion from morphisms to functions does not work, i.e., writing g a raises a "function expected" error. This error can be fixed by writing (g : X ⟶ Y) a.

References

• [F. Borceux, Handbook of Categorical Algebra 2][borceux-vol2]

In this section we prove that for every P there is an equivalence class that contains precisely all the zero morphisms ending in P and use this to define the zero pseudoelement.