Category theory MOC

Zero morphism

In a category 𝖢, a constant morphism 𝑐 ∈𝖢(𝑋,𝑌) satisfies 𝑐𝑓 =𝑐𝑔 for any 𝑓,𝑔 ∈𝖢(𝑍,𝑋) and 𝑍 ∈𝖢, whereas a coconstant morphism 𝑐 ∈𝖢(𝑋,𝑌) satisfies 𝑓𝑐 =𝑔𝑐 for any 𝑓,𝑔 ∈𝖢(𝑌,𝑍). A zero morphism is both a constant and coconstant morphism. #m/def/cat

A category 𝖢 is said to have zero morphisms iff for any two objects 𝑋,𝑌 ∈𝖢 there is a fixed morphism 0𝑋𝑌 ∈𝖢(𝑋,𝑌) such that the following diagram commutes #m/def/cat

for any 𝑋,𝑌 ∈𝖢 and 𝑓,𝑔 ∈𝖢(𝑋,𝑌).

Properties

• A category with zero morphisms allows one to define the Kernels and cokernels of morphisms.

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