Franklin's Notes

Types of morphisms

In category theory , the following are different types of morphisms:

These properties of morphisms interact with each other and with composition in the following ways:

Here are some specific examples of each of these in abstract categories (work in progress):

Here are some useful facts relating these types of morphisms to hom-sets . In a locally small category $\mathsf{C}$, we have that

In the category $\mathsf{Set}$, a morphism is an isomorphism iff it is both monic and epic. But this is not true in general, and this fact is easy to forget! As a reminder, here are several examples of categories with nonisomorphisms that are monic and epic:



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