## § Intuition for limits in category theory

#### § A characterization of limits

The theorem that characterizes limits is this:
A category has Finite Limits iff it has all finite products and equalizers. limits have maps out of them.

#### § Ravi Vakil's intuition for limits: Take sub things.

• An element of a limit gives one of each of its ingredients. For example,$K[[X]] = \lim_n \text{degree}~n~\text{polynomials}$, since we can geta degree n polynomial for all n, from any power series by truncation.
• limits have maps out of them.
• RAPL: right adjoints preserve limits.
• Limits commute with limits.
• These are mentioned in "Algebraic geometry in the time of Covid: Pseudolecture 2"
• Also, recalling that kernels are limits allows one to remember that we havemaps that go out of a limit.

#### § Limits are things you get maps out of.

• product has projections going out of it.
• gcd is less than the things that build it: gdb(a, b) -> a, b since gcd(a, b) < a and gcd(a, b) < b.
• single point set can be mapped out of.
• type Limit f = forall a. f a

{-# LANGUAGE ExplicitForAll #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE KindSignatures #-}