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Judging by Prop 3.1 of Lenses, fibrations and universal translations, lenses are just algebras for our possibility monad, at necessity and possibility.
I’m told this is good. But I don’t have access at the moment: https://dl.acm.org/citation.cfm?doid=3191697.3191718
and of course we should cite the paper by Mitchell that started the discussion.
Given
necessity comonad : possibility monad = jet comonad : infinitesimal disk bundle monad,
and since lens (in computer science) (at least lawlike ones) seem to be algebras for the possibility monad, what are algebras for the infinitesimal disk bundle?
We’d need, $E$ over $X$ and a map $T_{inf} X \times_X E \to E/X$.
Oh, is it just the constants $E = K \times X$. But then what’s that exciting about lenses?
That paper Lenses, fibrations and universal translations continues that when the $V$ of views is such that $V \to 1$ is split epi then a lens just is a projection.
So people look at more general situations?
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