
A self-contained, brief and complete formulation of Voevodsky's Univalence Axiom - jessup
http://www.cs.bham.ac.uk/~mhe/agda-new/UnivalenceFromScratch.html
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mbid
It is misleading to say that elementary toposes/zfc don't have something like
identity types. Every elementary topos is locally cartesian closed, and
locally cartesian closed categories are (equivalent to) models of extensional
Martin-Löf type theory, which is an extension of intensional type theory.
Identity types in extensional type theory are responsible for the existence of
equalizers in its model categories. Universes in type theory correspond to
inaccessible cardinals/Grothendieck universes in ZFC or object classifiers in
elementary toposes, at least informally (I doubt there is published work
here).

Thus, the term asserting the univalence axiom corresponds to a certain
morphism in an elementary topos with object classifiers. The point is, I
guess, that such a morphism exists only in the degenerate topos, i.e. the one
equivalent to the single object-single morphism category. Only in higher
toposes/categories can non-degenerate examples of the univalence axiom be
found.

It should also be noted that you can already identify isomorphic objects
("types") of 1-categories without much harm. Formally, if you have a
contractible groupoid G contained in a category C, then the quotient map C ->
C/G that collapses all objects of G onto a single object and all morphisms in
G onto the identity at that object is an equivalence of categories. This works
in particular if G is given by an isomorphism of distinct objects.

It still boggles my mind why type theorists think that "function
extensionality" and quotients, two entirely 1-categorical concepts, are best
treated using homotopy coherent diagrams. And it is unclear since when proving
theorems in _less_ generality (because additional axioms are assumed) is
considered progress.

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EtDybNuvCu
This makes me feel much better. While I think that MLTT and HoTT are valid
lines of maths inquiry on their own, for their own merits, I'm not sure what's
gained over the topos-based view of foundations.

In particular, the article claims that the Curry-Howard-Lambek correspondence
can be used to recover logical sentences from MLTT sentences. Then surely MLTT
universes each form a topos of some sort? I suppose that a few hours on nCat
would clear this up for me.

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pacala
Dilettante here: What's interesting about *TT is that people have built
mechanical theorem checkers [Coq, Agda, Lean, ...], and people are working on
building automated theorem provers based on those programatic foundations.

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EtDybNuvCu
Check out Metamath; we can do this for any serious foundations:
[http://us.metamath.org/downloads/metamath.pdf](http://us.metamath.org/downloads/metamath.pdf)

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pacala
Have somebody _done_ it for the more general foundations based on category
theory? If I understand correctly, Metamath is a set of tools enabling such a
formulation, but not the formulation itself. See ML vs. theorem prover
implemented in ML.

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mbid
There are some developments, for example "Globular":
[https://arxiv.org/abs/1612.01093](https://arxiv.org/abs/1612.01093)

I don't think there is a proof assistant that's really based on categorical
foundations. I'd love to see something like that though.

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Jeff_Brown
Is there any obvious application of this in software design?

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SolarNet
Type systems.

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Jeff_Brown
What will it let me represent that extant type systems don't?

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SolarNet
Nothing. However it will provide a consistent basis for translating a wider
variety of systems to existing type systems.

