Pseudoelements in abelian categories #
A pseudoelement of an object
X in an abelian category
C is an equivalence class of arrows
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
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.
Here are the metatheorems we provide:
- A morphism
fis zero if and only if it is the zero function on pseudoelements.
- A morphism
fis an epimorphism if and only if it is surjective on pseudoelements.
- A morphism
fis a monomorphism if and only if it is injective on pseudoelements if and only if
∀ a, f a = 0 → f = 0.
- A sequence
f, gof morphisms is exact if and only if
∀ a, g (f a) = 0and
∀ b, g b = 0 → ∃ a, f a = b.
fis a morphism and
a, a'are such that
f a = f a', then there is some pseudoelement
f a'' = 0and for every
g a' = 0 → g a = g a''. We can think of
a - a', but don't get too carried away by that: pseudoelements of an object do not form an abelian group.
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.
Implementation notes #
It appears that sometimes the coercion from morphisms to functions does not work, i.e.,
g a raises a "function expected" error. This error can be fixed by writing
(g : X ⟶ Y) a.
- [F. Borceux, Handbook of Categorical Algebra 2][borceux-vol2]
This is just composition of morphisms in
C. Another way to express this would be
(Over.map f).obj a, but our definition has nicer definitional properties.
f : X ⟶ P and
g : Y ⟶ P are called pseudo-equal if there is some object
R and epimorphisms
p : R ⟶ X and
q : R ⟶ Y such that
p ≫ f = q ≫ g.
Pseudoequality is transitive: Just take the pullback. The pullback morphisms will be epimorphisms since in an abelian category, pullbacks of epimorphisms are epimorphisms.
If two elements are pseudo-equal, then their composition with a morphism is, too.
f induces a function
pseudoApply f on pseudoelements.
A coercion from morphisms to functions on pseudoelements.
Applying a pseudoelement to a composition of morphisms is the same as composing with each morphism. Sadly, this is not a definitional equality, but at least it is true.
Composition of functions on pseudoelements is composition of morphisms.
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
The pseudoelement induced by an arrow is zero precisely when that arrow is zero.
The zero morphism maps every pseudoelement to 0.
An extensionality lemma for being the zero arrow.
A monomorphism is injective on pseudoelements.
A morphism that is injective on pseudoelements only maps the zero element to zero.
A morphism that only maps the zero pseudoelement to zero is a monomorphism.
An epimorphism is surjective on pseudoelements.
A morphism that is surjective on pseudoelements is an epimorphism.
Two morphisms in an exact sequence are exact on pseudoelements.
If two morphisms are exact on pseudoelements, they are exact.
If two pseudoelements
y have the same image under some morphism
f, then we can form
z. This pseudoelement has the properties that
f z = 0 and for all
g y = 0 then
g z = g x.
f : P ⟶ R and
g : Q ⟶ R are morphisms and
p : P and
q : Q are pseudoelements such
f p = g q, then there is some
s : pullback f g such that
fst s = p and
snd s = q.
Remark: Borceux claims that
s is unique, but this is false. See
Counterexamples/Pseudoelement.lean for details.
In the category
Module R, if
y are pseudoequal, then the range of the associated
morphisms is the same.