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I have added some new material to Boolean algebra and to ultrafilter. In the former, I coined the term ’unbiased Boolean algebra’ for the notion which describes Boolean algebras as equivalent to finite-product-preserving functors $Fin_+ \to Set$ from the category of finite nonempty sets, and the term $k$-biased Boolean algebra to refer to the multiplicity of ways in which Boolean algebras could be considered monadic over $Set$.
In ultrafilter, I added some material which gives a number of universal descriptions of the ultrafilter monad. This is in part inspired by some discussions I’m having with Tom Leinster, who remarked recently at the categories list that the ultrafilter monad could be described as a codensity monad. All this is related to the unbiased Boolean algebras and to the remarks due to Lawvere, which were described on an earlier revision; this material has been reworked.
Great to have you back, Todd!
(Thanks, Urs (-: ) I have recently learned through a Math Overflow discussion that the usual term for what I called “$k$-biased Boolean algebra” is a $k$-valued Post algebra (after Emil Post), so I inserted this information in Boolean algebra.
Awesome! I’m now checking how well this stuff works in constructive mathematics.
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