timelets

Categorical Algebra with Segal Conditions

Tue, 18 Feb 2025 19:48:37 GMT

Among other things, the ant solution to the grain sorting problem given to Psyche by Aphrodite can be modeled as a replacement of an Inert with an Active. The same applies to the Trasnsformer solution of the translation problem, etc.

upd. the Odysseus solution to the Sirens problem also fits the pattern

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Mon, 03 Feb 2025 07:31:21 GMT

I keep coming back to this video about the relationship between (pre-)sheafs and cohomology. Here he says that "the number one technique in mathematics is turning any problem into a linear algebra problem.

More generally, Lawvere often talks about mapping geometry to algebra.

https://youtu.be/RPuWHN0BTio?si=U0h7YM-3GlcyvnS5&t=1890

D --> J <-- T ( c: D --> T is the solution to a choice problem, per Lawvere).

d: D --> J
e: T --> J
c: D --> T

This diagram is a regular Kan extension problem, with a cohomology twist, i.e. assigning values to both objects and arrows.

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Sun, 01 Sep 2024 05:59:58 GMT

The word ‘_perceive_’ is, in our common usage, shot through and through
with the notion of cognitive apprehension. So is the word
‘_apprehension_’, even with the adjective _cognitive_ omitted. I will
use the word ‘_prehension_’ for _uncognitive apprehension_: by this I
mean _apprehension_ which may or or may not be cognitive.

Whitehead. Science..., 1925.

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If we modeled this prehension as a CT monad we could develop and algebra of prehension, with quantative thresholds that separate the uncognitive and cognitive.

https://ncatlab.org/nlab/show/state+monad

X -> [W, WxY]

Here the operation [W, Wx(-)) is the monad on the type system which is induced by the above adjunction; and this latter function is naturally regarded as a morphism in the Kleisli category of this monad.

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