## June 16, 2019

### Applied Category Theory Meeting at UCR

#### Posted by John Baez

The American Mathematical Society is having their Fall Western meeting here at U. C. Riverside during the weekend of November 9th and 10th, 2019. Joe Moeller and I are organizing a session on Applied Category Theory!

## June 13, 2019

### What’s a One-Object Sesquicategory?

#### Posted by John Baez

A **sesquicategory**, or $1\frac{1}{2}$-category, is like a 2-category, but without the interchange law relating vertical and horizontal composition of 2-morphisms:

$(\alpha \cdot \beta)(\gamma \cdot \delta) = (\alpha \gamma) \cdot (\beta \delta)$

Better, sesquicategories are categories enriched over $(Cat,\square)$: the category of categories with its “white” tensor product. In the *cartesian* product of categories $C$ and $D$, namely $C \times D$, we have the law

and we can define $f \times g$ to be either of these. In the *white* tensor product $C \square D$ we do not have this law, and $f \times g$ makes no sense.

What’s a one-object sesquicategory?

## June 5, 2019

### Nonstandard Models of Arithmetic

#### Posted by John Baez

A nice quote:

There seems to be a murky abyss lurking at the bottom of mathematics. While in many ways we cannot hope to reach solid ground, mathematicians have built impressive ladders that let us explore the depths of this abyss and marvel at the limits and at the power of mathematical reasoning at the same time.

This is from Matthew Katz and Jan Reimann’s nice little book *An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics*. I’ve been been talking to my old friend Michael Weiss about nonstandard models of Peano arithmetic on his blog. We just got into a bit of Ramsey theory. But you might like the whole series of conversations.

## June 3, 2019

### Why Category Theory Matters

#### Posted by John Baez

No, I’m not going to tell you why category theory matters. To learn that, you must go here:

- Robb Seaton, Why category theory matters, rs.io.

## May 28, 2019

### A Question on Left Adjoints

#### Posted by John Baez

*guest post by Jade Master*

I’m interested in internalizing the “free category on a reflexive graph” construction.

We can define reflexive graphs internal to any category $C$, and categories internal to $C$ whenever $C$ has finite limits. Suppose $C$ has finite limits; let $\mathsf{RGph}(C)$ be the category of reflexive graphs internal to $C$, and let $\mathsf{Cat}(C)$ be the category of categories internal to $C$. There’s a forgetful functor

$U \colon \mathsf{Cat}(C) \to \mathsf{RGraph}(C)$

When does this have a left adjoint?

I’m hoping it does whenever $C$ is the category of algebras of a Lawvere theory in $\mathsf{Set}$, but I wouldn’t be surprised if it were true more generally.

Also, I’d really like references to results that answer my question!

## May 23, 2019

### Polyadic Boolean Algebras

#### Posted by John Baez

I’m getting a bit deeper into model theory thanks to some fun conversations with my old pal Michael Weiss… but I’m yearning for a more category-theoretic approach to classical first-order logic. It’s annoying how in the traditional approach we have theories, which are presented syntatically, and models of theories, which tend to involve some fixed set called the domain or ‘universe’. This is less flexible than Lawvere’s approach, where we fix a doctrine (for example a 2-category of categories of some sort), and then say a theory $A$ and a ‘context’ $B$ are both objects in this doctrine, while a model is a morphism $f: A \to B.$

One advantage of Lawvere’s approach is that a theory and a context are clearly two things of the same sort — that is, two objects in the same category, or 2-category. This means we can think not only about models $f : A \to B$, but also models $g : B \to C$, so we can compose these and get models $g f : A \to C$. The ordinary approach to first-order logic doesn’t make this easy.

So how can we update the apparatus of classical first-order logic to accomplish this, without significantly changing its content? Please don’t tell me to use intuitionistic logic or topos theory or homotopy type theory. I love ‘em, but today I just want a 21st-century framework in which I can state the famous results of classical first-order logic, like Gödel’s completeness theorem, or the compactness theorem, or the Löwenheim–Skolem theorem.

## May 20, 2019

### Young Diagrams and Schur Functors

#### Posted by John Baez

What would you do if someone told you to invent something a lot like the natural numbers, but even cooler? A tough challenge!

I’d recommend ‘Young diagrams’.

## May 16, 2019

### Partial Evaluations

#### Posted by John Baez

*guest post by Martin Lundfall and Brandon Shapiro*

This is the third post of Applied Category Theory School 2019.

In this blog post, we will be sharing some insights from the paper Monads, partial evaluations and rewriting by Tobias Fritz and Paolo Perrone.

## April 29, 2019

### Right Properness of Left Bousfield Localizations

#### Posted by Mike Shulman

*(Guest post by Raffael Stenzel)*

This post is a sequel to the discussion of the mysterious nature of right properness and its understanding as an instance of coherence problems for presenting $(\infty,1)$-categorical structure. The last post discussed a relation between right properness of a model category $\mathcal{M}$ and locally cartesian closedness of the underlying $(\infty,1)$-category $\mathrm{Ho}_{\infty}(\mathcal{M})$. While the two properties – that is right properness of $\mathcal{M}$ on the one hand and locally cartesian closedness of $\mathrm{Ho}_{\infty}(\mathcal{M})$ on the other – are generally independent of each other, the post and its subsequent discussion basically established an equivalence of the two properties in the context of Cisinski model categories in the following sense; a presentable $(\infty,1)$-category $\mathcal{C}$ is locally cartesian closed iff there is a right proper Cisinski model category $\mathcal{M}$ whose underlying $(\infty,1)$-category is equivalent to $\mathcal{C}$. In this follow up, we aim to generalize this connection, and we do so via replacing “locally cartesian closedness” of $(\infty,1)$-categories by “semi-left exactness” of their reflective localizations.

While this is not meant to be an exhaustive description of the nature of right properness either, it hopefully gives another stimulus to kindle further discussion.

## April 28, 2019

### Generalized Petri Nets

#### Posted by John Baez

*guest post by Jade Master*

I just finished a paper which uses Lawvere theories to generalize Petri nets. I can think of two reasons why people might be interested in this:

Category theorists love Lawvere theories and are in awe of their power. However, it can be hard to find instances where Lawvere theories are used to get something specific and practical accomplished.

There are lots of papers on Petri nets and their variants. The bibliography on Petri nets world has over 8500 citations. This generalization puts some of the more popular variants under a common framework and allows for exploration of the relationships between them.

## April 24, 2019

### Twisted Cohomotopy Implies M-Theory Anomaly Cancellation

#### Posted by David Corfield

The latest instalment of Urs’s march towards M-theory is out on the arXiv today, Twisted Cohomotopy implies M-Theory anomaly cancellation, (arXiv:1904.10207).

A review of the program to date appears in the recent:

- Domenico Fiorenza, Hisham Sati, Urs Schreiber, The rational higher structure of M-theory.

In the latest paper the authors are moving beyond the approximation of *rational* cohomology to explain from first principles the folklore about anomaly cancellations in M-theory. The generalized non-abelian cohomology theory known as cohomotopy theory, and in particular its twisted version, appears to have all the answers, accounting for six conditions on the cocycles corresponding to the C-field.

## April 19, 2019

### Can 1+1 Have More Than Two Points?

#### Posted by John Baez

I feel I’ve asked this before… but now I *really* want to know. Christian Williams and I are working on cartesian closed categories, and this is a gaping hole in my knowledge.

**Question 1.** Is there a cartesian closed category with finite coproducts such that there exist more than two morphisms from $1$ to $1 + 1$?

Cartesian closed categories with finite coproducts are a nice context for ‘categorified arithmetic’, since they have $0$, $1$, addition, multiplication and exponentiation. The example we all know is the category of finite sets. But *every* cartesian closed category with finite coproducts obeys what Tarski called the ‘high school algebra’ axioms:

$x + y \cong y + x$

$(x + y) + z \cong x + (y + z)$

$x \times 1 \cong x$

$x \times y \cong y \times x$

$(x \times y) \times z \cong x \times (y \times z)$

$x \times (y + z) \cong x \times y + x \times z$

$1^x \cong 1$

$x^1 \cong x$

$x^{(y + z)} \cong x y \times x z$

$(x \times y)^z \cong x^z \times y^z$

$(x^y)^z \cong x^{(y \times z)}$

together with some axioms involving $0$ which for some reason Tarski omitted: perhaps he was scared to admit that in this game we want $0^0 = 1$.

So, one way to think about my question is: *how weird can such a category be?*

## April 17, 2019

### Univalence in (∞,1)-toposes

#### Posted by Mike Shulman

It’s been believed for a long time that homotopy type theory should be an “internal logic” for $(\infty,1)$-toposes, in the same way that higher-order logic (or extensional type theory) is for 1-toposes. Over the past decade a lot of steps have been taken towards realizing this vision, but one important gap that’s remained is the construction of sufficiently strict universe objects in model categories presenting $(\infty,1)$-toposes: the object classifiers of an $(\infty,1)$-topos correspond directly only to a kind of “weakly Tarski” universe in type theory, which would probably be tedious to use in practice (no one has ever seriously tried).

Yesterday I posted a preprint that closes this gap (in the cases of most interest), by constructing strict univalent universe objects in a class of model categories that suffice to present all Grothendieck $(\infty,1)$-toposes. The model categories are, perhaps not very surprisingly, left exact localizations of injective model structures on simplicial presheaves, which were previously known to model all the rest of type theory; the main novelty is a new more explicit “algebraic” characterization of the injective fibrations, enabling the construction of universes.

## April 14, 2019

### The ZX-Calculus for Stabilizer Quantum Mechanics

#### Posted by John Baez

*guest post by Fatimah Ahmadi and John van de Wetering*

This is the second post of Applied Category Theory School 2019. We present Backens’ completeness proof for the ZX-calculus for stabilizer quantum mechanics.

## April 9, 2019

### Postdoctoral Researcher Position in Lisbon

#### Posted by John Huerta

Applications are invited for a postdoctoral researcher position in the “Higher Structures and Applications” research team, funded by the Portuguese funding body FCT.

For more, read on!