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Stream: learning: questions

Topic: Extensionality

dan pittman (Jul 24 2020 at 02:25):

Help me understand this?
https://twitter.com/andrejbauer/status/1286336084067209221?s=19

Yoneda : Category theory = Extensionality : Set theory

- Andrej Bauer (@andrejbauer)

I think I understand the Cayley analogy given in the replies, but don't totally understand what that has to do with extensionality.

Is it because the equivalent permutation subgroup is equal by extensionality?

Cole Comfort (Jul 24 2020 at 02:44):

dan pittman said:

Help me understand this?
https://twitter.com/andrejbauer/status/1286336084067209221?s=19

I think I understand the Cayley analogy given in the replies, but don't totally understand what that has to do with extensionality.

Is it because the equivalent permutation subgroup is equal by extensionality?

In both cases, things are the same if and only if they look the same externally.

dan pittman (Jul 24 2020 at 02:57):

That makes sense. I think at this point I've come to believe that an extensional definition of extensionality is undecidable :)

Jules Hedges (Jul 24 2020 at 08:55):

Cole Comfort said:

In both cases, things are the same if and only if they look the same externally.

I wonder if this is everything and it's basically a massive handwave (which would be the default option for a tweet), or if there's some more detail that Andrej secretly knows

Fabrizio Genovese (Jul 24 2020 at 09:54):

dan pittman said:

Help me understand this?
https://twitter.com/andrejbauer/status/1286336084067209221?s=19

I think I understand the Cayley analogy given in the replies, but don't totally understand what that has to do with extensionality.

Is it because the equivalent permutation subgroup is equal by extensionality?

Given an object $A$ in a category, a generalized element of $A$ is a morphism $X \to A$. $Hom(-,A)$ is then the set of all generalized elements of $A$. Yoneda says, among other things, that $A \simeq B$ iff $Hom(-,A) \simeq Hom(-,B)$. That is, two objects are isomorphic iff their generalized elements are. This is akin to "Two sets are equal if they have the same elements", sets are as extensional as you can get, hence... :slight_smile:

John Baez (Jul 24 2020 at 14:46):

Spelling it out a bit more carefully:

Traditional extensionality: two sets are equal iff they have the same elements.

Category-theoretic extensionality: two sets are isomorphic iff they have naturally isomorphic set of maps from a 1-element set into them.

Yoneda: Two objects are isomorphic iff they have naturally isomorphic sets of map from all other objects into them.

It's a special feature of the category of sets (and some other categories) that we don't need to map all other objects into them to get Yoneda to work: the terminal object 1 is enough.

John Baez (Jul 24 2020 at 14:48):

A category with this property is called well-pointed.

dan pittman (Jul 24 2020 at 16:21):

Fabrizio Genovese said:

Given an object $A$ in a category, a generalized element of $A$ is a morphism $X \to A$. $Hom(-,A)$ is then the set of all generalized elements of $A$. Yoneda says, among other things, that $A \simeq B$ iff $Hom(-,A) \simeq Hom(-,B)$. That is, two objects are isomorphic iff their generalized elements are. This is akin to "Two sets are equal if they have the same elements", sets are as extensional as you can get, hence... :)

Thanks for this excellent explanation; it makes perfect sense, and is, I think, how I was seeing it via Cayley.