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Stream: learning: questions

Topic: Product Preserving Functor

Cecilia Campani (Apr 01 2024 at 13:58):

Hello everyone! I have (probably) a simple question to make: do product preserving functor commutes with projections? I.e., if F: C \rightarrow D is a product preserving functor, axb is the product of a, b obects in C, and p_a and p_b are the projections, does it hold that F(p_a)=P_F(a) and F(p_b)=P_F(b)? I tried to prove it but without succes, I think F must be full and surjective on objects but I'm still not sure about it.

Morgan Rogers (he/him) (Apr 01 2024 at 14:06):

Hi! In brief, yes it's true that the projections are preserved. To give you the most helpful explanation of why, could you tell me what definition of product-preserving functor you are working with?
(PS. you can use double dollar signs for inline equations e.g. $F(p_a)$.)

Morgan Rogers (he/him) (Apr 01 2024 at 14:09):

Beware that an object can be a product of two objects in several ways. That is, the projections are part of the data of the product, and another choice of projections can sometimes be possible making the same object a product in a different way. Products are online unique up to unique isomorphism. So it doesn't necessarily make sense to demand that $F$ sends given product projections to "the" product projections from the image of $F$; rather you need to check that the images $F(p_a), F(p_b)$ present $F(a \times b)$ as a product.

Cecilia Campani (Apr 01 2024 at 14:34):

Yes, I'm working with the definition of n-Lab: https://ncatlab.org/nlab/show/product-preserving+functor.

Cecilia Campani (Apr 01 2024 at 14:35):

Thank you so much for the help, also for the P.s.!

Cecilia Campani (Apr 01 2024 at 14:36):

So, you mean that I have to check if the two maps satisfy the universal property of the product, right?

Ralph Sarkis (Apr 01 2024 at 14:49):

If you don't know $F$ preserves products, then yes, you need to show $F\pi_X$ and $F\pi_Y$ satisfy the universal property of the product $FX \times FY$, and if you can do that for all $X$ and $Y$ that have a product, then you can conclude $F$ preserves products.
If you know $F$ preserves products, then you also know (by definition) that $F\pi_X$ and $F\pi_Y$ satisfy the universal property.

It is a good exercise to come up with a functor that satisfies $F(X\times Y) = FX \times FY$, but yet does not preserve the projections (hence does not preserve products).

Cecilia Campani (Apr 01 2024 at 16:55):

Eventually, I proved what I needed to! But thanks for the help, really appreciated!