# Contrasting isomorphisms, equivalencies, and adjunctions

01 Jan 2020 ⇐ Notes archive(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)

When I was first introduced to adjunctions, I reacted in the way Spivak anticipated during an applied category theory lecture (timestamped, transcribed below).

…and when people see this definition [of an adjunction], they think “well, that seems…fine. I’m glad you told me about…that.”

And it wasn’t until I stumbled upon a Catsters lecture from 2007 (!) where Eugenia Cheng clarified the intent behind the definition (and contrasted it to isomorphisms and equivalencies).

To start, assume we have two categories `C`

and `D`

with functors, `F`

and `G`

between them (`F`

moving from `C`

to `D`

and `G`

in the other direction).

There are a few possible scenarios we could find ourselves in.

- Taking the round trip—i.e.
`GF`

and`FG`

—is**equal**to the identity functors on`C`

and`D`

(denoted with`1_C`

and`1_D`

below). - The round trip is
**isomorphic**to each identity functor. - Or, the round trip lands us
**a morphism away**from where we started.

Moving between the scenarios, there’s a sort of “relaxing” of strictness:

- The round trip is the identity functor.
- […] an isomorphism away from the identity functor.
- […] a hop away from identity functor.