32,163 research outputs found
Sets in homotopy type theory
Homotopy Type Theory may be seen as an internal language for the
-category of weak -groupoids which in particular models the
univalence axiom. Voevodsky proposes this language for weak -groupoids
as a new foundation for mathematics called the Univalent Foundations of
Mathematics. It includes the sets as weak -groupoids with contractible
connected components, and thereby it includes (much of) the traditional set
theoretical foundations as a special case. We thus wonder whether those
`discrete' groupoids do in fact form a (predicative) topos. More generally,
homotopy type theory is conjectured to be the internal language of `elementary'
-toposes. We prove that sets in homotopy type theory form a -pretopos. This is similar to the fact that the -truncation of an
-topos is a topos. We show that both a subobject classifier and a
-object classifier are available for the type theoretical universe of sets.
However, both of these are large and moreover, the -object classifier for
sets is a function between -types (i.e. groupoids) rather than between sets.
Assuming an impredicative propositional resizing rule we may render the
subobject classifier small and then we actually obtain a topos of sets
The real projective spaces in homotopy type theory
Homotopy type theory is a version of Martin-L\"of type theory taking
advantage of its homotopical models. In particular, we can use and construct
objects of homotopy theory and reason about them using higher inductive types.
In this article, we construct the real projective spaces, key players in
homotopy theory, as certain higher inductive types in homotopy type theory. The
classical definition of RP(n), as the quotient space identifying antipodal
points of the n-sphere, does not translate directly to homotopy type theory.
Instead, we define RP(n) by induction on n simultaneously with its tautological
bundle of 2-element sets. As the base case, we take RP(-1) to be the empty
type. In the inductive step, we take RP(n+1) to be the mapping cone of the
projection map of the tautological bundle of RP(n), and we use its universal
property and the univalence axiom to define the tautological bundle on RP(n+1).
By showing that the total space of the tautological bundle of RP(n) is the
n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an
(n+1)-cell attached to it. The infinite dimensional real projective space,
defined as the sequential colimit of the RP(n) with the canonical inclusion
maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises
as the subtype of the universe consisting of 2-element types. Indeed, the
infinite dimensional projective space classifies the 0-sphere bundles, which
one can think of as synthetic line bundles.
These constructions in homotopy type theory further illustrate the utility of
homotopy type theory, including the interplay of type theoretic and homotopy
theoretic ideas.Comment: 8 pages, to appear in proceedings of LICS 201
The Seifert-van Kampen Theorem in Homotopy Type Theory
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding "synthetic homotopy theory". Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the "encode-decode" method, enforced homotopy-invariance, and lack of underlying sets
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