Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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

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

When does this have a left adjoint?

I’m hoping it does whenever CC is the category of algebras of a Lawvere theory in Set\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!

Posted at 6:54 AM UTC | Permalink | Followups (29)

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 AA and a ‘context’ BB are both objects in this doctrine, while a model is a morphism f:AB.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:ABf : A \to B, but also models g:BCg : B \to C, so we can compose these and get models gf:ACg 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.

Posted at 10:56 PM UTC | Permalink | Followups (35)

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’.

Layer 1

Posted at 12:01 AM UTC | Permalink | Followups (8)

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.

Posted at 9:54 PM UTC | Permalink | Followups (1)