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Stream: learning: questions

Topic: Universal Mapping Properties as Representable Functors


view this post on Zulip Suraaj K S (Feb 27 2025 at 01:34):

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:

  1. An Initial object in a category
  2. A Representation of a Functor into Set
  3. A Universal Arrow

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?

view this post on Zulip Josselin Poiret (Feb 27 2025 at 09:31):

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 that x : 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.

view this post on Zulip Josselin Poiret (Feb 27 2025 at 09:33):

also, notice that maps in Set NA ℕ → A don't exactly classify pairs a:A a : A with AA A → A , you don't necessarily get that the original map is defined by induction on a "global" endo-function

view this post on Zulip Josselin Poiret (Feb 27 2025 at 09:41):

this is very similar to the ability of defining some objects by their impredicative encodings: take the end X(AX)(BX)X \int_X (A → X) → (B → X) → X in Set (impredicative, or just take the index category to be Set<κ \mathrm{Set}_{<κ} big enough to contain A×B A × B ), this gives you the product A×B A × B . 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

view this post on Zulip Morgan Rogers (he/him) (Feb 27 2025 at 14:39):

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.

view this post on Zulip Suraaj K S (Feb 27 2025 at 19:14):

Ah I see, perhaps it is safe to say that the link that I shared is slightly misleading..

view this post on Zulip Suraaj K S (Feb 27 2025 at 19:18):

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'?

view this post on Zulip Josselin Poiret (Feb 28 2025 at 06:53):

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 AB A ⊗ B as the representing object for the functor XBilin(A,B;X) X ↦ \mathrm{Bilin}(A, B; X)

view this post on Zulip Mike Shulman (Mar 02 2025 at 16:43):

However, there is at least a connection. By the Yoneda lemma, a representation of a functor F:CSetF:C\to \rm Set is equivalent to a universal arrow from 1Set1\in \rm Set to FF, which is equivalently an initial object of the comma category (1/F)(1/F). An initial object of a category is equivalently a limit of its identity functor, and if FF preserves limits (a necessary condition for it to be representable) then limits in (1/F)(1/F) are created in CC. Therefore, if FF preserves limits and CC has a limit of the forgetful functor (1/F)C(1/F) \to C, that limit gives a representation of FF. And if FF is accessible or otherwise small-determined, one can often reduce this to a limit of a small diagram. (This is essentially a β\beta-reduction of the adjoint functor theorem, using the fact that a representation of FF is equivalent to a value of a left adjoint to FF at 1Set1\in \rm Set.)