In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a zero morphism is a special kind of
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
, 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 ''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 has a unique "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 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)