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Prikvačeni tweet
stuff next to my bed: - pill bottles - fidget spinner - copy of Capital vol 1 - copy of Topoi: The Categorial Analysis of Logic - duct tape
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i really didn't think anything ray sipe did could surprise me at this point buthttps://twitter.com/raysipe/status/1223448212566171649 …
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one stray thought: if we only use the address ranges as the underlying p.c.m. instead of heaps w/ contents included, then maybe we get a semantics for a pure language which captures part of the notion that the runtime can update in place if the program is linear
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which sounds like a pretty good notion of "separating product" to me! just not sure how to handle mutation yet :)
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the action of the functor on morphisms is just inclusions. then an element of (A ⊗ B)(h) works out to be a pair of a contiguous region in h storing ints and a tree in h, such that none of the locations involved in the tree are the same as the contiguous region!
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so for example, maybe R is a p.c.m. of heaps w/ the extension/divisibility ordering, A could be integer arrays, B could be trees. given a heap h, A(h) is the set of contiguous regions in h storing integers, B(h) is similar in spirit but more complicated.
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the quotient is that if y_1 ≤ y_1' and y_2 ≤ y_2', with y_1, y_2 and y_1', y_2' both pairs as above, then an A(y_1) × B(y_2) pair is identified with its A(y_1') × B(y_2') weakening as an element of (A ⊗ B)(y).
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so, up to a quotient, a y-owning value of A ⊗ B is a pair of a y_1-owning A and a y_2-owning B, such that y_1 ⊗ y_2 is defined and ≤ y.
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now say A, B in PSh(R^op). we can take a "separating product" of A and B, the day convolution. writing ⊗_R for the promonoidal product, (A ⊗ B)(y) = ∫^{y_1, y_2} (y_1 ⊗_R y_2)(y) × A(y_1) × B(y_2)
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if we have x ≤ y, then we can go F(x) → F(y)—we can weaken our bound on what a given value owns. this lines up well with the fact that having nontrivial inequalities in an ordered p.c.m. can cause the resulting separation logic to be affine, whereas discrete order = linear
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say F is an object of PSh(R^op). let's think of it as semantics for a type. given x in R, we'll think of F(x) as the set of values of type F which own x, or at least for which x is an upper bound (in R's ordering) on their resource footprint.
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yes, I think it might be! say R is a preordered p.c.m. of "resources". the p.c.m. structure furnishes a promonoidal structure. take PSh(R^op); we can set ⊗ and ⊸ to be day convolution and its adjoint.
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fun 2 work out why they call PSh(N) the "topos of trees" (with N a thin category under its usual ordering)
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john baez posts to the ncatlab about optics, its a sign that my time is about to come
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sarahzrf proslijedio/la je Tweet
Like I always say, regret narratives are popular because they're emotionally resonant with cis people - because they think *they* would regret it - because they would - because they are cis.https://twitter.com/TransActualUK/status/1181230218251558918 …
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sarahzrf proslijedio/la je TweetHvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi
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