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. #m/thm/cat

Proof

Let $𝐹 : 𝖢 \to 𝖣$ and $𝑓 \in 𝖢 (𝑋, 𝑌)$ be an isomorphism. Then there exists $𝑓 - 1 \in 𝖢 (𝑌, 𝑋)$ such that $𝑓 - 1 𝑓 = i d 𝑋$ and $𝑓 𝑓 - 1 = i d 𝑌$ . It follows that $(𝐹 𝑓 - 1) (𝐹 𝑓) = 𝐹 i d 𝑋 = i d 𝐹 𝑋$ and $(𝐹 𝑓) (𝐹 𝑓 - 1) = (𝐹 i d 𝑌) = i d 𝐹 𝑌$ . Therefore $𝐹 𝑓 \in 𝖣 (𝐹 𝑋, 𝐹 𝑌)$ has inverse $𝐹 𝑓 - 1 \in 𝖣 (𝐹 𝑌, 𝐹 𝑋)$ , whence $𝐹 𝑓$ 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 $𝐹 : 𝖳 𝗈 𝗉 \to 𝖢$ assigns to any topological space is immediately a Topological property.¹

Fully faithful

If a functor $𝐹$ is Fully faithful functor this becomes bidirectional:

𝑋 ≅ 𝑌 ⟺ 𝐹 𝑋 ≅ 𝐹 𝑌

#state/tidy | #lang/en | #SemBr

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

Functors encode invariants of isomorphism classes

Fully faithful

Footnotes