iiinova — where your digital self keeps evolving

AI in math is like high-level programming: it changes the unit of thought. Proofs become components in larger structures, freeing attention for how theories fit together. The goal is understanding; AI adds abstraction. The concern is its use for manipulation, dominance, and oppression.

010

Conor Sheridan @qucat.bsky.social · 14/09/2026

Nine and a half months after applying for a PhD, Glasgow has finally resolved the application. Rejected due to “the high level of competition.” Good to have closure on one of 2026’s longer-running open problems.

000

Conor Sheridan @qucat.bsky.social · 14/09/2026

I would very much like to work on designing and implementing a type theory capable of verifying proofs and constructions in spectral algebraic geometry. What would be needed for such a thing? Intuitively something like a dependent differential linear homotopy type theory?

000

Conor Sheridan @qucat.bsky.social · 12/09/2026

I’ve put up a paper version of some recent work on non-Cartesian internal categories and distributors. The main idea is a poly-virtual calculus that lets the relevant category and distributor operations be defined without assuming cotensors exist. tinyurl.com/y95ynfyn

tinyurl.com

Formal-categories-in-poly-virtual-equipments/Formal_categories_and_distributors__Non_Cartesian_Internal_Categories.pdf at main · uzum4ke/Formal-categories-in-poly-virtual-equipments

Contribute to uzum4ke/Formal-categories-in-poly-virtual-equipments development by creating an account on GitHub.

000

Conor Sheridan @qucat.bsky.social · 11/09/2026

This weeks essay became very long so I've split it into a 5 part series. Here is this weeks post on part I: Lawvere theories as spaces of operations , the next post will be available around this time next Friday! tinyurl.com/4kve5rfj

tinyurl.com

Geometry from Algebra I

Lawvere theories as spaces of operations

000

Conor Sheridan @qucat.bsky.social · 09/09/2026

You all know that problems and proofs aren't a finite resource, right?

000

Conor Sheridan @qucat.bsky.social · 08/09/2026

Is there a generalisation of the noncartesian internal categories of Aguilar and the generalized multicategories of Shulman&Crutwell? They don't seem to play very nicely together, or is this just my imagination?

000

Conor Sheridan @qucat.bsky.social · 08/09/2026

People should have a legally protected right to use AI without surrendering the plaintext of their conversations to AI providers. Cryptographically verifiable confidential inference needs to become the required standard as the technology matures.

000

Conor Sheridan @qucat.bsky.social · 02/09/2026

In order to improve my writing skills I've started a blog! I’m aiming to write one expository essay a week. The first is Spectra from a Logical Perspective, you can see it at either of the following links: Substack: tinyurl.com/z979axay PDF: tinyurl.com/3bnke68c

tinyurl.com

Spectra from a Logical Perspective

What happens if equality is allowed to carry more information than a simple yes-or-no answer?

000

Conor Sheridan @qucat.bsky.social · 01/09/2026

My old notes on fuzzy logic have evolved, first into a higher enriched form and via stabilization into a spectral information geometry! I have two more chapters to finish: Blackwell information and thermodynamics, then I'll release the notes as a milestone here on social media.

010

Conor Sheridan @qucat.bsky.social · 01/09/2026

Besides theoretical computer science, I’ve had a long-term interest in the geometry of physics. I’ve now put the first public version of that project on GitHub. If category-theoretic geometry interests you: tinyurl.com/ds8jnhbs

tinyurl.com

Towards-a-Brave-New-Variational-Geometry/Towards_a_Brave_New_Variational_Geometry: Foundations.pdf at main · uzum4ke/Towards-a-Brave-New-Variational-Geometry

Contribute to uzum4ke/Towards-a-Brave-New-Variational-Geometry development by creating an account on GitHub.

010

Conor Sheridan @qucat.bsky.social · 26/08/2026

What is the categorification of a Turing machine?

000

Conor Sheridan @qucat.bsky.social · 19/08/2026

Species/symmetric sequences have a natural bosonic Fock calculus; Kapranov links supergeometry to stable homotopy. Has anyone combined these, e.g. by stabilizing the species calculus, to obtain a genuinely fermionic/super Fock structure?

000

Conor Sheridan @qucat.bsky.social · 19/08/2026

Cool observation: one can build a bicategory whose objects are free word monoids over alphabets and whose morphisms are Turing transducers. A Turing machine then appears as a specialization of a guarded trace between a Moore machine (tape) and a Mealy machine (controller).

001

Conor Sheridan @qucat.bsky.social · 15/08/2026

Formally speaking: Given a type system and an operational semantics, what more is needed for a 'meaning explanation'?

000

Conor Sheridan @qucat.bsky.social · 14/08/2026

Looking for non-trivial examples of virtual double categories arising in TCS or logic, ideally many-object examples where the virtuality has a concrete computational/logical meaning.

011

Conor Sheridan @qucat.bsky.social · 09/08/2026

I was reading the Day-Street paper on Quantum Categories over the weekend and in the introduction they mention a hope of interaction between linear logic and quantum group theory, but I haven't been able to find much progress on this, does anyone know of any references on this?

000

Reposted by Conor Sheridan

Zanzi Tangle @zanzi.bsky.social · 06/08/2026

after two years of development, I finally demoed Jermaine for the first time - a language with fully symmetric data and codata, and native call-cc.

5447

Conor Sheridan @qucat.bsky.social · 03/08/2026

For the past few months, I’ve been working on a personal research project, *Categorical Automata Theory*. The first volume, *Residual Behaviour and Operational Algebra*, is still a work in progress, but it’s now at a stage I’m happy to share: github.com/uzum4ke/Auto...

github.com

000

Conor Sheridan @qucat.bsky.social · 31/07/2026

A rather unexpected connection between two projects: While studying higher-order GSOS and bialgebraic semantics as a mathematical account of compilation, I began wondering whether the same machinery could provide a genuinely intensional denotational semantics for SF-calculus.

100

Conor Sheridan @qucat.bsky.social · 25/07/2026

In favourable settings, a linear exponential modality admits a free differential completion, yielding a model of differential linear logic. Operads and related algebraic theories also admit deformation theory. Is there a translation between these constructions?

010