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Stream: theory: mathematics

Topic: Linearly distributive Isbell envelopes

Mike Shulman (Aug 24 2025 at 18:17):

In another thread Matteo Capucci (he/him) said about the extended real numbers $[-\infty,\infty]$:

in https://arxiv.org/abs/2406.04936 I, like you, observed that the 'two additions', distinguished by how they resolve the indeterminate form $-\infty+\infty$, come together to make the extended reals an (isomix) $*$-autonomous quantale.

That is a cute example of a $\ast$-autonomous category! I don't think I've seen that before. (It's unfortunate that (in linear logic terminology and notation) the "positive" addition is the one with $(-\infty)\otimes (+\infty) = -\infty$ and the "negative" addition is the one with $(-\infty)\mathbin{⅋}(+\infty) = +\infty$, but c'est la vie.)

The extended reals are the [[MacNeille completion]] of the rationals, i.e. the posetal saturated [[Isbell envelope]]. If I'm not mistaken, its two additions are the result of extending the addition on $\mathbb{Q}$ to its presheaf and copresheaf categories, respectively, and then to the saturated Isbell envelope by acting on one or the other component and recovering the other by saturation. But this is something one could do for the saturated Isbell envelope of any monoidal category. Is the result always $\ast$-autonomous? Or at least linearly distributive?

John Baez (Aug 24 2025 at 20:14):

Digressing a bit:

@Owen Lynch, @Joe Moeller and I have a paper where we give $[-\infty,+\infty]$ the structure of a convex space, extending $\mathbb{R}$ with its usual convex space structure, such that any nontrivial convex combination of $-\infty$ and $+\infty$ is $-\infty$. There's another choice where any nontrivial convex combination of $-\infty$ and $+\infty$ is $+\infty$, but we need our choice to prove some theorems about entropy!

Matteo Capucci (he/him) (Sep 05 2025 at 06:22):

Thanks Mike! Yeah notation (and terminology) is quite unfortunate for this example :sweat_smile:

Matteo Capucci (he/him) (Sep 05 2025 at 06:24):

The two additions are indeed the two ways of extending that on the rationals as infima or suprema preserving ones. I'm not familiar enough with Isbell envelopes to really help with your other question though.
Maybe you can say what 'posetal' (though I can guess) and 'saturated' mean?

Mike Shulman (Sep 05 2025 at 06:38):

By "posetal" I mean "2-enriched". By "saturated" I mean the objects satisfying "Isbell duality" in the sense of https://ncatlab.org/nlab/show/Isbell%20envelope#isbell_duality.

Mike Shulman (Sep 05 2025 at 06:44):

I may have been wrong about saying you could do this (meaning define the two tensors) for any saturated Isbell envelope of a monoidal category, though. My suggested definition for $\otimes$ acts on F-saturated objects, while my definition of $⅋$ acts on P-saturated objects, but even when applied to Isbell-duality objects it's not clear that either tensor product produces another Isbell-duality object.