David Roberts scratch 2015

This page is intended to be a dumping ground for thoughts. No guarantee is made for correctness. On the other hand, I would be happy to discuss stuff from here if you have ideas. See David Roberts for my contact details. See also: scratch 2014, scratch 2016, scratch 2017

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2015

November

3

Get from a bounded geometric morphism $p:E \to F$ of bounded toposes an internal category in $F$ such that $E$ is sheaves on that. Consult the Elephant. If $E$ is already given by sheaves on an external site, then we can take for a bound for $p$ the coproduct of the representables.

1

A forcing notion that is a poset is $\kappa$-closed (or rather $\lt \kappa$-closed), for $\kappa$ a regular cardinal, if it has $\lambda$-codirected limits for all $\lambda \lt \kappa$; or better, $\lambda$-filtered limits. In the usual definition, one takes $\lambda$-sequential limits, or at least weak $\lambda$-sequential limits (a lower bound without being an infimum). But in a poset, (weak) filtered limits are the same as sequential limits.

However, this property is only used to show that $\kappa$-closed posets become $\kappa$-distributive? after forcing with a poset that is $\kappa$-cc (i.e. satisfies the $\kappa$-chain condition).

Recall that $\kappa$-distributive means that the intersection of less than $\kappa$ covering sieves is again a covering sieve. For posets in traditional forcing, covering sieves are identified with dense sets of elements. It is a theorem in classical logic that $\kappa$-closed implies $\kappa$-distributive (i.e. the existence of sequential limits of the appropriate size implies the intersection of that many covering sieves is a covering sieve). But for more general sites I don’t see how this helps.

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October

24

Lee Rudolph A Morse function for a surface in $\mathbb{CP}^3$, Topology 1975, Vol.14(4):301–303, doi:10.1016/0040-9383(75)90013-0

Suggested at this MO answer to my question “Vector field on a K3 surface with 24 zeroes”.