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Delta-Epsilon Definition of Limit Reminds Me of Adjoint Functors

I am still mastering the concept of adjoint functors. I found it very helpful to think of them as "approximate inverses", in the case of preorders, from above as least upper bounds, and from below as greatest lower bounds.

Yesterday I was recalling the delta-epsilon definition of a limit L of a function f(x) at a point x=c:

For every ε>0, there exists a δ such that, for all x ∈ D, if 0 < |x-c| < δ, then |f(x)-L| < ε.

What struck me is that we can think of δ as a function δ(ε), in which case it becomes apparent that δ(ε) is constructed in the opposite direction as f(x). For simplicity, let us consider the case where c=0 and L=0. Let X be the preorder of real numbers on (0,1], and Y likewise. Define functors F:X→Y and δ:Y→X. Then F and δ are similar to adjoint functors because:

If |x|<δ(ε), then |f(x)| < ε.

If we had: |x|<δ(ε) iff |f(x)| < ε

then I think F and δ would be adjoint functors because, in a preorder, there is at most a single morphism ≤, so then we could say:

$$ \textrm{hom}_X(\delta (\varepsilon),x) \cong \textrm{hom}_Y(\varepsilon,F(x)) $$ So I suppose δ(ε) would be the adjoint of F if it was the optimal δ. I'm curious what that would mean. But also it seems that typically in analysis, a suboptimal δ(ε) works just fine. I wonder why this example isn't discussed more often, especially given that category theory seems distant to many analysts. Thank you for correcting any misunderstandings I may have.

Comments

  • 1.

    I suppose that category theory can shift the matter from constructing an explicit δ(ε) to considering whether the optimal δ(ε) exists. If F has an adjoint functor, then the optimal δ(ε) exists, and a sufficient δ(ε) exists, so F is continuous. And if F has no adjoint functor, then an optimal δ(ε) does not exist, but then no sufficient δ(ε) exists, so F is not continuous. Am I arguing correctly?

    Comment Source:I suppose that category theory can shift the matter from constructing an explicit δ(ε) to considering whether the optimal δ(ε) exists. If F has an adjoint functor, then the optimal δ(ε) exists, and a sufficient δ(ε) exists, so F is continuous. And if F has no adjoint functor, then an optimal δ(ε) does not exist, but then no sufficient δ(ε) exists, so F is not continuous. Am I arguing correctly?
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