category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a zero morphism is a special kind of

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

exhibiting properties like the morphisms to and from a

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

Definitions

Suppose C is a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

, 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

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

''W'' in C and any , ''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 0_''AB'' : ''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 0_''XY'' necessarily are zero morphisms and form a compatible system of zero morphisms. If C is a category with zero morphisms, then the collection of 0_''XY'' 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 In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

has a ″zero morphism", then the category "has zero morphisms".

Examples

Related concepts

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 Mor_C(''X'', ''Y''). Thus, every category with a zero object is a category with zero morphisms given by the composition 0_''XY'' : ''X'' → 0 → ''Y''. If a category has zero morphisms, then one can define the notions of

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...

and

cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...

for any morphism in that category.

References

*Section 1.7 of * .

Notes

{{Reflist Morphisms 0 (number)