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August 29, 2025

Equivalence via Surjections

Posted by Tom Leinster

Pick a type of categorical structure: say bicategories, or monoidal categories, or whatever you like. Some of the functors between structures are equivalences, in whatever the appropriate sense might be. And some of those equivalences have one or both of these two properties:

  • They’re not just essentially surjective in every dimension — they’re actually surjective in every dimension.

  • They don’t just preserve the structure up to isomorphism or equivalence — they strictly preserve it.

Call an equivalence with both these properties a strict surjective equivalence. So a strict surjective equivalence is an equivalence of a very special and easy kind.

General principle: the standard notion of equivalence between structures is generated by just these very special ones. For example, two bicategories are biequivalent if and only if they can be linked up by a zigzag of strict surjective equivalences.

Why should we care? Because there are some types of structure where the right notion of equivalence isn’t clear, and this principle guides us to it. For example, it tells us the right notion of equivalence for double categories.

All this is done in my new paper:

Tom Leinster, Equivalence via surjections. arXiv:2508.20555, 2025.

Continue reading “Equivalence via Surjections” …
Posted at 10:23 PM UTC | Permalink | Followups (27)

August 28, 2025

Burrito Monads, Arrow Kitchens, and Freyd Category Recipes

Posted by Tom Leinster

Guest post by Khyathi Komalan and Andrew Krenz

From Lawvere’s Hegelian taco to Baez’s layer cake analogy to Eugenia Cheng’s How to Bake Pi, categorists have cultivated a rich tradition of culinary metaphors and similes. A well-known example in the world of computation is Mark Dominus’s “monads are like burritos” — where a tortilla (computational context) wraps diverse ingredients (values) to create a cohesive entity (effectful value) whose burrito structure is maintained as the meal moves down the assembly line (undergoes computations).

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August 14, 2025

Safeguarded AI Meeting

Posted by John Baez

This week, 50 category theorists and software engineers working on “safeguarded AI” are meeting in Bristol. They’re being funded by £59 million from ARIA, the UK’s Advanced Research and Invention Agency.

The basic idea is to develop a mathematical box that can contain a powerful genie. More precisely:

By combining scientific world models and mathematical proofs we will aim to construct a ‘gatekeeper’, an AI system tasked with understanding and reducing the risks of other AI agents. In doing so we’ll develop quantitative safety guarantees for AI in the way we have come to expect for nuclear power and passenger aviation.

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Posted at 12:06 PM UTC | Permalink | Followups (10)

August 5, 2025

(BT) Diversity from (LC) Diversity

Posted by Tom Leinster

Guest post by Mark Meckes

Around 2010, in papers that both appeared in print in 2012, two different mathematical notions were introduced and given the name “diversity”.

One, introduced by Tom Leinster and Christina Cobbold, is already familiar to regular readers of this blog. Say XX is a finite set, and for each x,y∈Xx,y \in X we have a number Z(x,y)=Z(y,x)∈[0,1]Z(x,y) = Z(y,x) \in [0,1] that specifies how “similar” xx and yy are. (Typically we also assume Z(x,x)=1Z(x,x) = 1.) Fix a parameter q∈[0,∞]q \in [0,\infty]. If pp is a probability distribution on XX, then the quantity D q Z(p)=(∑ x∈supp(p)(∑ y∈supp(p)Z(x,y)p(y)) q−1p(x)) 1/(1−q) D_q^Z(p) = \left(\sum_{x\in supp(p)} \left( \sum_{y\in supp(p)} Z(x,y) p(y)\right)^{q-1} p(x)\right)^{1/(1-q)} (with the cases q=1,∞q=1,\infty defined by taking limits) can be interpreted as the “effective number of points” in XX, taking into account both the similarities between points as quantified by ZZ and the weights specified by pp. Its logarithm logD q Z(p)\log D_q^Z(p) is a refinement of the qq-Rényi entropy of pp. The main motivating example is when XX is a set of species of organisms present in an ecosystem, and D q Z(p)D_q^Z(p) quantifies the “effective number of species” in XX, accounting for both similarities between species and their relative abundances. This family of quantities turns out to subsume many of the diversity measures previously introduced in the theoretical ecology literature, and they are now often referred to as Leinster–Cobbold diversities.

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Posted at 4:34 PM UTC | Permalink | Followups (4)

August 2, 2025

Jack Morava

Posted by John Baez

Today I heard from David Benson that Jack Morava died yesterday. This comes as such a huge shock that I can’t help but hope Benson was somehow misinformed. Morava has been posting comments to the n-Café and sending emails to me even very recently.

This is all I know, now.

Posted at 12:22 PM UTC | Permalink | Followups (2)