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Stream: learning: questions

Topic: Extending "Functors" that act on a subset of Set

Davi Sales Barreira (Mar 17 2023 at 16:01):

Hello, friends. I'm trying to translate the concepts of Category Theory to Julia programming. In Julia, we have strcutcs which are containers that can be turned into a functor by defining an fmap on it. We can make structs parametric, e.g.

struct F{T} where T
 a::T
end
fmap(f, A::F) = F(f(A.a))
F(f::Function) = x::F -> fmap(f,x)

Thus, we have defined a functor F that takes types T to F{T}, and the morphism mapping is given by the fmap.
Now, sometimes we define a struct in which the subtype is fixed:

struct F****
 a::String
end
fmap(f, A::F) = F(f(A.a))

Hence, our F is not a functor in this case, since it is not properly defined for every possible object (type). Is there a canonical extension of this type of "semi"-functor? I mean, if we have a "semi-endofunctor" $F:\mathbf{Set} \to \mathbf{Set}$ where it is actually defined only in a subset of $\mathbf{Set}$. Is there a canonical extension for such thing? Is there even a name for this stuff?

Josselin Poiret (Mar 17 2023 at 16:18):

Here F (the second one) is just a type isomorphic to String. You're pretty far from a functor, you've just defined one object in the category. Basically, if F is your functor above (the first definition), you computed F(String)

Spencer Breiner (Mar 17 2023 at 16:22):

It's worth noting that you can think of a fixed type as a functor.

The type itself is most naturally represented as a functor from the terminal category ${\tt String}:{\bf 1\to Type}$ (rather than a functor $\bf Type \to Type$).

But then you can compose with the unique map to get back a "constant" functor $\bf Type \to 1 \to Type$

Josselin Poiret (Mar 17 2023 at 16:23):

iiuc that composite would be written

struct F{T} where T
 a::String
end

although i don't know anything about julia

Spencer Breiner (Mar 17 2023 at 16:24):

That looks right to me

Spencer Breiner (Mar 17 2023 at 16:25):

F**** would be the functor $\bf 1\to Type$, i.e., just another type.

Ralph Sarkis (Mar 17 2023 at 16:26):

Spencer Breiner said:

But then you can compose with the unique map to get back a "constant" functor $\bf Type \to 1 \to Type$

But this sends any functions on strings to the identity function, which I guess is not what Davi is defining. The functor should send any function between strings to itself.

Spencer Breiner (Mar 17 2023 at 16:32):

I like Julia a lot, but I do get frustrated by the lack of types on the functions.

I suppose what you're describing would be the inclusion of the endomorphism monoid of String, and the implicit typing on says it should accept a string A.a and return a string (the input to F).

Spencer Breiner (Mar 17 2023 at 16:33):

You can still map constantly through the identity function, but you're right that this is not the fmap that Davi defined.