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Lyes Saadi 2026-01-15 17:57:21 +01:00
parent 08f2664cec
commit 82d6d2f9b9
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2.typ
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@ -809,3 +809,39 @@ where $bb(L)_!(A,B) := bb(L)(!A, B)$
&g << f = f compose !f compose nu_A
$
]
#exercise[
Prove this defines a category :
- Identity is $epsilon$
- $f << epsilon_A = f compose !epsilon_A compose eta_A = f$
- $epsilon_B << f = epsilon_B compose !f compose eta_A = f compose epsilon_(!A) compose eta_A = f$
- $
h << (g << f) &= h compose !(g compose !f compose nu_A) compose nu_A = h compose !g compose !!f compose !nu_A compose nu_A \
(h << g) << f &= (h compose !g compose nu_B) compose !f compose nu_A = h compose !g compose !!f compose nu_(!A) compose nu_A
$
Axioms of comonads
// https://q.uiver.app/#r=typst&q=WzAsNCxbMSwwLCIhQSJdLFsxLDEsIiEhQSJdLFswLDEsIiFBIl0sWzIsMSwiIUEiXSxbMCwyLCJpZF8oIUEpIiwyXSxbMCwzLCJpZF8oIUEpIl0sWzEsMiwiIWVwc2lsb25fQSJdLFsxLDMsImVwc2lsb25fKCFBKSIsMl0sWzAsMSwibnVfQSIsMl1d
- #align(center, diagram({
node((0, 0), [$!A$])
node((0, 1), [$!!A$])
node((-1, 1), [$!A$])
node((1, 1), [$!A$])
edge((0, 0), (-1, 1), [$id_(!A)$], label-side: right, "->")
edge((0, 0), (1, 1), [$id_(!A)$], label-side: left, "->")
edge((0, 1), (-1, 1), [$!epsilon_A$], label-side: left, "->")
edge((0, 1), (1, 1), [$epsilon_(!A)$], label-side: right, "->")
edge((0, 0), (0, 1), [$nu_A$], label-side: right, "->")
}))
// https://q.uiver.app/#r=typst&q=WzAsNCxbMCwwLCIhQSJdLFsxLDAsIiEhQSJdLFsxLDEsIiEhIUEiXSxbMCwxLCIhIUEiXSxbMCwxLCJudV9BIl0sWzEsMiwibnVfKCFBKSJdLFszLDIsIiFudV9BIiwyXSxbMCwzLCJudV9BIiwyXV0=
- #align(center, diagram({
node((0, 0), [$!A$])
node((1, 0), [$!!A$])
node((1, 1), [$!!!A$])
node((0, 1), [$!!A$])
edge((0, 0), (1, 0), [$nu_A$], label-side: left, "->")
edge((1, 0), (1, 1), [$nu_(!A)$], label-side: left, "->")
edge((0, 1), (1, 1), [$!nu_A$], label-side: right, "->")
edge((0, 0), (0, 1), [$nu_A$], label-side: right, "->")
}))
]