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October 21, 2015

When Not To Use ‘The’

Posted by David Corfield

I’ve revised A Note on ‘The’ and ‘The Structure of’ in Homotopy Type Theory, which we discussed a few months ago – The Structure of A.

As Mike said back then, trying to define ‘structure of ’ in HoTT is a form of ‘noodling around’, and I rather think that working up a definition of ‘the’ is more important. The claim in the note is that we should only form a term ‘The AA’ for a type AA, if we have established the contractibility of AA. I claim that this makes sense of types which are singleton sets, as well as the application of ‘the’ in cases where category theorists see universal properties, such as ‘the product of…’.

Going down the hh-levels, contractible propositions are true ones. I think it’s not too much of a stretch to see the ‘the’ of ‘the fact that PP’ as an indication of the same principle.

But what of higher hh-levels? Is it the case that we don’t, or shouldn’t, use ‘the’ with types which are non-contractible groupoids? One case that came to mind is with algebraic closures of fields. Although people do say ‘the algebraic closure of a field FF’ since any two such are isomorphic, as André Henriques writes here, a warning is often felt necessary about the use of ‘the’ in that these isomorphisms are not canonical. Do people here also get a little nervous with ‘the universal cover of a space’? Perhaps intuitively one provides a little extra structure (map in or map out, say) which makes the isotropy trivial.

I was also wondering if we see traces of this phenomena in natural language, but I think the examples I’m coming up with (the way to hang a symmetrical painting, the left of a pair of identical socks) are better thought of as concerning the formation of terms in equivariant contexts (as at nLab: infinity-action), and the subject of a lengthy discussion a while ago on coloured balls.

Posted at 11:16 AM UTC | Permalink | Followups (32)

October 20, 2015

Four Tribes of Mathematicians

Posted by John Baez

Since category theorists love to talk about their peculiar role in the mathematics community, I thought you’d enjoy this blog article by David Mumford, which discusses four “tribes” of mathematicians with different motivations. I’ll quote just a bit, just to whet your appetite for the whole article:

The title refers to an “astonishing experimental investigation” of what your brain is doing when you experience mathematical beauty. This was carried out here:

But on to the four tribes….

Posted at 1:06 AM UTC | Permalink | Followups (5)

October 5, 2015

Configurations of Lines and Models of Lie Algebras (Part 2)

Posted by John Baez

To get deeper into this paper:

we should think about the 24-cell, the D 4\mathrm{D}_4 Dynkin diagram, and the Lie algebra 𝔰𝔬(8)\mathfrak{so}(8) that it describes. After all, its this stuff that underlies the octonions, which in turn underlies the exceptional Lie algebras, which are the main subject of Manivel’s paper.

Posted at 10:03 PM UTC | Permalink | Followups (23)

October 4, 2015

Configurations of Lines and Models of Lie Algebras (Part 1)

Posted by John Baez

I’m really enjoying this article, so I’d like to talk about it here at the n-Category Café:

It’s a bit intense, so it may take a series of posts, but let me just get started…

Posted at 11:06 PM UTC | Permalink | Followups (7)