Functor

Functors encode invariants of isomorphism classes

Since functors take compositions to compositions and identities to identities, they also take isomorphisms to isomorphisms, thereby preserving isomorphism classes. cat

Proof

Let and be an isomorphism. Then there exists such that and . It follows that and . Therefore has inverse , whence is an isomorphism.

This is a fundamental idea that captures the very essence of what makes category theory useful. For example, in Topology MOC, the value an arbitrary functor assigns to any topological space is immediately a Topological property.¹

Fully faithful

If a functor is Fully faithful functor this becomes bidirectional:

---

tidy | en | sembr

Footnotes

1. 2020, Topology: A categorical approach, p. 11 ↩