### The Structure of A

#### Posted by David Corfield

I attended a workshop last week down in Bristol organised by James Ladyman and Stuart Presnell, as part of their Homotopy Type Theory project.

Urs was there, showing everyone his magical conjuring trick where the world emerges out of the opposition between **$\emptyset$** and **$\ast\;$** in Modern Physics formalized in Modal Homotopy Type Theory.

Jamie Vicary spoke on the Categorified Heisenberg Algebra. (See also John’s page.) After the talk, interesting connections were discussed with dependent linear type theory and tangent (infinity, 1)-toposes. It seems that André Joyal and colleagues are working on the latter. This should link up with Urs’s Quantization via Linear homotopy types at some stage.

As for me, I was speaking on the subject of my chapter for the book that Mike’s *Introduction to Synthetic Mathematics* and John’s *Concepts of Sameness* will appear in. It’s on reviving the philosophy of geometry through the (synthetic) approach of cohesion.

In the talk I mentioned the outcome of some further thinking about how to treat the phrase ‘the structure of $A$’ for a mathematical entity. It occurred to me to combine what I wrote in that discussion we once had on The covariance of coloured balls with the analysis of ‘the’ from The King of France thread. After the event I thought I’d write out a note explaining this point of view, and it can be found here. Thanks to Mike and Urs for suggestions and comments.

The long and the short of it is that there’s no great need for the word ‘structure’ when using homotopy type theory. If anyone has any thoughts, I’d like to hear them.

Posted at April 12, 2015 2:01 PM UTC
## Re: The Structure of A

Nice! (Nit: “Introduction to Synthetic Mathematics” isn’t the title of my chapter, just a blog post excerpted from a first draft of it — not much of which may make it into the final version at all.)

In section 3 you say

but I don’t see an immediate justification of that claim in the note. I suppose section 4 could be read as a sort of justification, but it’d be interesting to see if this can be justified on its own, prior to applying “the” to it.

One general principle (often honored in the breach) is that for a type to be called $foo$, it ought to be the case that “$x:foo$” means “$x$ is a foo”. This convention is in force when calling the universe “$Type$”, and also for examples like $List(A)$ and $Equiv(A,B)$. So if we want to pronounce $\sum_{X:U} Equiv(A,X)$ as “structure of $A$”, it ought to be the case that $x:\sum_{X:U} Equiv(A,X)$ means “$x$ is a structure of $A$”. That’s not

quitegrammatically correct, but I think a sufficiently generous reader could interpret it as meaning that $x$ is a structure of the same sort as $A$, or that $x$ has exactly the same structure that $A$ does. Which seems roughly correct: if $x = (B,f)$ then the equivalence $f$ allows us to transport any structure possessed by $A$ over to $B$ in a canonical way.Did you have an argument like this in mind?