Cokernel

of a morphism in a category

The concept dual to the concept of the kernel of a morphism in a category. In categories of vector spaces, groups, rings, etc. it describes a largest quotient object of an object $ B $ that annihilates the image of a homomorphism $ \alpha : A \rightarrow B $.

Let $ \mathfrak K $ be a category with null morphisms. A morphism $ \nu : B \rightarrow C $ is called a cokernel of a morphism $ \alpha : A \rightarrow B $ if $ \alpha \nu = 0 $ and if any morphism $ \phi $ such that $ \alpha \phi = 0 $ can be expressed in unique way as $ \phi = \nu \psi $. A cokernel of a morphism $ \alpha $ is denoted by $ \mathop{\rm coker} \alpha $.

If $ \nu = \mathop{\rm coker} \alpha $ and $ \nu ^ \prime = \mathop{\rm coker} \alpha $ then $ \nu ^ \prime = \nu \xi $ for a unique isomorphism $ \xi $.

Conversely, if $ \nu = \mathop{\rm coker} \alpha $ and $ \xi $ is an isomorphism, then $ \nu ^ \prime = \nu \xi $ is a cokernel of $ \alpha $. Thus, all cokernels of a morphism $ \alpha $ form a quotient object of $ B $, which is denoted by $ \mathop{\rm Coker} \alpha $. If $ \nu = \mathop{\rm coker} \alpha $, then $ \nu $ is a normal epimorphism. The converse need not be true. The cokernel of the zero morphism $ 0: A \rightarrow B $ is $ 1 _ {B} $. The cokernel of the unit morphism $ 1 _ {A} $ exists if and only if $ \mathfrak K $ contains a zero object.

In a category $ \mathfrak K $ with a zero object, a morphism $ \alpha : A \rightarrow B $ has a cokernel if and only if $ \mathfrak K $ contains a co-Cartesian square with respect to the morphisms $ \alpha $ and $ 0: A \rightarrow 0 $. This condition is satisfied, in particular, for any morphism of a right locally small category with a zero object and products.

Comments

The co-Cartesian square, or fibred sum or pushout, of two morphisms $ f: S \rightarrow A $, $ g: S \rightarrow B $ is (if it exists) a commutative diagram

$$ \begin{array}{rcc} S & \stackrel{f}{\rightarrow} & A \\ { g } \downarrow &{} & \downarrow { {f _ {1} } } \\ B & \stackrel{g_1}{\rightarrow} &B \amalg _ {S} A \\ \end{array} $$

such that for any two morphisms $ a: A \rightarrow Y $, $ b: B \rightarrow Y $ such that $ af = bg $ there exists a unique morphism $ h: B \amalg _ {S} A \rightarrow Y $ for which $ a = hf _ {1} $, $ b = hg _ {1} $.