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Stream: learning: questions

Topic: Universal Mapping Properties as Representable Functors

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:

An Initial object in a category
A Representation of a Functor into Set
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?

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.

Josselin Poiret (Feb 27 2025 at 09:33):

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

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 $\int_X (A → X) → (B → X) → X$ in Set (impredicative, or just take the index category to be $\mathrm{Set}_{<κ}$ big enough to contain $A × B$ ), this gives you the product $A × B$ . 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

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.

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..

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'?

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

Mike Shulman (Mar 02 2025 at 16:43):

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