## Varieties of [B|M]-sets (part 2)

### Richard Garner – 15 June 2022

Last year I gave some talks describing how (finitary) cartesian closed varieties can be identified with pairs of a Boolean algebra B and a monoid M which act on each other in a certain way: I call this a matched pair [B|M].
In this talk we get into the nitty gritty of this, by describing the variety to which such a matched pair [B|M] corresponds; this is the titular variety of [B|M]-sets. We will see explicitly why [B|M]-sets form a cartesian closed category, and describe simple conditions under which they form a topos; and under which the forgetful functor from [B|M]-sets to [B|1]-sets is a closed functor.

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