I don't understand the difference between creating and reflecting limits in category theory as defined on p. 151 in Peter Smith's introductory textbook. Theorems 95 and 96 seem to confirm this but apparently creating is a stronger condition than reflecting. Please help me out
SQT
Leo Parker
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Wyatt Brown
Why do you brainlets not even know how to ask stupid questions? At least post theorems 95 and 96, brainlet
Christian Hall
Theorem 95. If the functor [math] F: \mathbf{C} \to \mathbf{D}[/math] creates limits of shape [math] \mathsf{J} [/math], it reflects them.
Theorem 96. Suppose [math] F: \mathbf{C} \to \mathbf{D}[/math] is a functor, that [math]\mathbf{D}[/math] has limits of shape [math]\mathsf{J}[/math] and [math]F[/math] creates such limits. Then [math]\mathbf{C}[/math] has limits of shape [math]\mathsf{J}[/math] anf [math]F[/math] preserves them.
Henry Brooks
Huh, while writing this I think I got it and I can't believe I missed it before.
Creation implies reflection and preservation, I guess.
Austin Gutierrez
some people (nlab) even take the definition of creating a limit as saying the diagram J has a limits whenever FJ does, and F preserves and reflects the limit of J.
reflecting a limit only means that if L is a cone on J, and FL is a limit of FJ, then L was already a limit of J. That is, if FL is a limit in the codomain, it was in the domain.
Creation means that if L is a limit of J, then FL must be a limit of FJ, and conversely, if FJ has a limit L, then J has a limit M and FM is isomorphic to L.
Christian Jenkins
What's the difference between these two statements?
[math]\forall x\exists y\exists z [P(x) \to Q(yz)] [/math]
and
[math]\forall xP(x) \to \exists y\exists zQ(yz) [/math]
And if there is no difference, is it true that in general you can just move quantifiers to the outermost scope? Why don't we just always move all quantifiers to the outermost/widest scope if that's the case?
Jason Russell
How do I show that image of a homomorphism isn't necessarily a normal subgroup? I want to do it without any explicit examples.
David Sanchez
I personally found an inclusion of a non-normal subgroup [math]H[/math] of [math]G[/math] to be an example ([math]\phi: H \hookrightarrow G[/math]) but this is so dissatisfying because of concreteness
Aaron Morales
someone answer this shit I was always wondering if the order really matters and im trash at logic
Grayson Bailey