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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 , and categories internal to whenever has finite limits. Suppose has finite limits; let be the category of reflexive graphs internal to , and let be the category of categories internal to . There’s a forgetful functor
When does this have a left adjoint?
I’m hoping it does whenever is the category of algebras of a Lawvere theory in , 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 syntactically, 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 and a ‘context’ are both objects in this doctrine, while a model is a morphism
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 , but also models , so we can compose these and get models . 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 1
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.