Abstract
It is difficult to overestimate the importance of modularity for specifying and reasoning about software [1], or for checking large and complex mathematical arguments [8,9,10]. The goal of this presentation is to explain in what way a recent development in type theory, the formulation of the axiom of univalence, addresses these modularity issues.
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- 1.
One important fact will be that, in contrast with set theory, all statements expressible in type theory are transportable.
- 2.
To give another (maybe more surprising) example: the collection of all linear orders with a fixed finite number of elements is a large collection, but it has no non trivial automorphisms, and should be considered to be the
as the groupoid with one object and only the identity morphism. - 3.
We write \(\varPi (x~y:A)B\) for \(\varPi (x:A)\varPi (y:A)B\).
- 4.
For a trivial, but significant [7] example, the center of group is automatically invariant by any automorphism of a group, and in particular, it is a normal subgroup.
as the groupoid with one object and only the identity morphism.References
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Coquand, T. (2017). Type Theory and Formalisation of Mathematics. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_1
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