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This (https://www.hedonisticlearning.com/posts/styles-of-category-theory.html), if I understand correctly, gives three ways of defining what a universal mapping property is:
Set
I was thinking a little bit about 2 - Is it the case that if I want to define an object in a category C
having a universal mapping property, I just need to 'encode this property' as a functor F (covariant / contravariant) into Set
, and just define the object to be the representation of F?
For instance, if P
,Q
are objects in a category C
, I can define their product to be the representation of the functor x : C ↦ (C(x,P) * C(x,Q)) : Set
.
As another example, I wanted to try to define the natural number object in a category as a representation of a functor into Set
. Is this possible? I thought that x : C ↦ C(1,x) * C(x,x)
might work, but this isn't even a functor I think? What am I missing?
Suraaj K S said:
As another example, I wanted to try to define the natural number object in a category as a representation of a functor into
Set
. Is this possible? I thought thatx : C ↦ C(1,x) * C(x,x)
might work, but this isn't even a functor I think? What am I missing?
The natural number object doesn't have a purely co- or contravariant property, so cannot be defined in this way unfortunately.
also, notice that maps in Set don't exactly classify pairs with , you don't necessarily get that the original map is defined by induction on a "global" endo-function
this is very similar to the ability of defining some objects by their impredicative encodings: take the end in Set (impredicative, or just take the index category to be big enough to contain ), this gives you the product . The same can't be said for the general impredicative encoding of ℕ, one needs to use some form of parametricity, like with reflexive graph categories
Although a natural number object is initial in a category of coproduct diagrams of a particular form. The obstruction is that it's not simply a limit/colimit in the category where it lives.
Ah I see, perhaps it is safe to say that the link that I shared is slightly misleading..
I think we can say that every limit 'is just' a representation of a functor. Is the converse true as well? Can we think of representations of functors (into Set) as 'just limits'?
Suraaj K S said:
I think we can say that every limit 'is just' a representation of a functor. Is the converse true as well? Can we think of representations of functors (into Set) as 'just limits'?
you can usually write way more functors from a specific category into set than what is permissible using just the language of category theory. Take for example the definition of the tensor product of vector spaces as the representing object for the functor
However, there is at least a connection. By the Yoneda lemma, a representation of a functor is equivalent to a universal arrow from to , which is equivalently an initial object of the comma category . An initial object of a category is equivalently a limit of its identity functor, and if preserves limits (a necessary condition for it to be representable) then limits in are created in . Therefore, if preserves limits and has a limit of the forgetful functor , that limit gives a representation of . And if is accessible or otherwise small-determined, one can often reduce this to a limit of a small diagram. (This is essentially a -reduction of the adjoint functor theorem, using the fact that a representation of is equivalent to a value of a left adjoint to at .)