Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Zero morphism
Summary
In , a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
Suppose C is a , and f : X → Y is a morphism in C. The morphism f is called a constant morphism (or sometimes left zero morphism) if for any W in C and any g, h : W → X, fg = fh. Dually, f is called a coconstant morphism (or sometimes right zero morphism) if for any object Z in C and any g, h : Y → Z, gf = hf. A zero morphism is one that is both a constant morphism and a coconstant morphism.
A category with zero morphisms is one where, for every two objects A and B in C, there is a fixed morphism 0AB : A → B, and this collection of morphisms is such that for all objects X, Y, Z in C and all morphisms f : Y → Z, g : X → Y, the following diagram commutes:
The morphisms 0XY necessarily are zero morphisms and form a compatible system of zero morphisms.
If C is a category with zero morphisms, then the collection of 0XY is unique.
This way of defining a "zero morphism" and the phrase "a category with zero morphisms" separately is unfortunate, but if each hom-set has a ′′zero morphism", then the category "has zero morphisms".
If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f : X → 0 and g : 0 → Y. Then, gf is a zero morphism in MorC(X, Y). Thus, every category with a zero object is a category with zero morphisms given by the composition 0XY : X → 0 → Y.
If a category has zero morphisms, then one can define the notions of and cokernel for any morphism in that category.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.