jessicata comments on Why’s equality in logic less flexible than in category theory?

[jessicata]

The thing you have said (presence of an isomorphism) is not equality in category theory. Set-theoretic equality is equality in category theory (assuming doing category theory with set-theoretic foundations). Like, we could consider a (small) category as set of objects + set of morphisms + function assigning ordered pair of objects to each morphism.

Rather, what you’re talking about is a certain type of equivalence relation (presence of an isomorphism). It doesn’t always behave like equality, because it is not equality.

[Algon]

Thank you, that is clearly correct and I’m not sure why I made that error. Perhaps because equivalence seems more interesting in category theory than in set theory? Which is interesting. Why is equivalence more central in category theory than set theory?

[jessicata]

I think with category theory, isomorphism is the obvious equivalence relation on objects in a category, whereas in set theory, which equivalence relation to use depends on context. E.g. we could consider reals as equivalence classes of Cauchy sequences of naturals (equivalent when their difference converges to 0). The equivalence relation here is explicit, it’s not like in category theory where it follows from other structures straightforwardly.

[Algon]

Besides isomorphisms and equality of objects, do category theorists use other notions of “equality”?

[Mateusz Bagiński]

There’s the equivalence of categories. Two categories are equivalent when they are isomorphic up to an isomorphism. Specifically, if you have two functors $F:C\to D$ and $G:D\to C$, such that there are natural isomorphisms (invertible natural transformations) $\alpha :F\circ G\stackrel{\sim }{\to }{1}_D$ and $\beta :1_C\stackrel{\sim }{\to }G\circ F$. On objects, this means that if you start at an object $X$ in $C$, then you can go to $FX$ in $D$ and then to $GFX$ in $C$ that is isomorphic to $X$: $GFX\cong X$. Similarly if you start at some object in $D$.

Equivalence is an isomorphism when the isomorphisms $GFX\cong X$ (equivalently,^[1] the natural transformations between the compositions and identity functors) are equalities.

An even weaker equality-like notion is adjunction and this where things start to get asymmetrical. There’s a few equivalent^[2] ways of defining them^[3], but the contextually simplest way (if not completely rigorous), since I just described equivalences, is that it’s an equivalence, except the natural transformations $\alpha :F\circ G\to 1_D$ and $\beta :1_C\to G\circ F$ are not (in general) isomorphic. So, you go $X\stackrel{F}{\to }FX\stackrel{G}{\to }GFX$ and you could instead have gotten there via some morphism $X\to GFX$ in the category $C$ but you may not be able to go back $GFX\to X$. On the other hand, starting at some $Y$ in $D$, you can take the trip $Y\stackrel{G}{\to }GY\stackrel{F}{\to }FGY$ and go back to where you started via some morphism within $D$, $FGY\to Y$. Again, there may not be a morphism $Y\to FGY$.

Then we say that F is left adjoint to G (equivalently, G is right adjoint to F), denoted $F⊣G$. The natural transformations ${\eta }_X:X\to GFX$ and ${ϵ}_Y:FGY\to Y$ are called the unit and the counit of the adjunction, respectively.

Seven Sketches in Compositionality introduces adjunctions in a slow and palatable way that is good for building an intuition, starting with Galois connections, which are just adjunctions for preorders, which are just Bool-categories.

1. ^

   no pun intended

2. ^

   double no pun intended

3. ^

   many such cases in category theory

[Algon]

Two categories are equivalent when they are isomorphic up to an isomorphism

Thaaaaat is a confusing sentence. But thankfully, the rest of your comment clears things up.

Why do you want this notion of equivalence or adjunction, rather than the stricter notion of isomorphism of categories?