## 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.