Kraus, Nicolai
(2015)
Truncation levels in homotopy type theory.
PhD thesis, University of Nottingham.
Abstract
Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type theory. We present several original results in homotopy type theory which are related to the truncation level of types, a concept due to Voevodsky.
To begin, we give a few simple criteria for determining whether a type is 0truncated (a set), inspired by a wellknown theorem by Hedberg, and these criteria are then generalised to arbitrary n. This naturally leads to a discussion of functions that are weakly constant, i.e. map any two inputs to equal outputs.
A weakly constant function does in general not factor through the propositional truncation of its domain, something that one could expect if the function really did not depend on its input. However, the factorisation is always possible for weakly constant endofunctions, which makes it possible to define a propositional notion of anonymous existence. We additionally find a few other nontrivial special cases in which the factorisation works. Further, we present a couple of constructions which are only possible with the judgmental computation rule for the truncation. Among these is an invertibility puzzle that seemingly inverts the canonical map from Nat to the truncation of Nat, which is perhaps surprising as the latter type is equivalent to the unit type.
A further result is the construction of strict ntypes in MartinLof type theory with a hierarchy of univalent universes (and without higher inductive types), and a proof that the universe U(n) is not ntruncated. This solves a hitherto open problem of the 2012/13 special year program on Univalent Foundations at the Institute for Advanced Study (Princeton).
The main result of this thesis is a generalised universal property of the propositional truncation, using a construction of coherently constant functions. We show that the type of such coherently constant functions between types A and B, which can be seen as the type of natural transformations between two diagrams over the simplex category without degeneracies (i.e. finite nonempty sets and strictly increasing functions), is equivalent to the type of functions with the truncation of A as domain and B as codomain. In the general case, the definition of natural transformations between such diagrams requires an infinite tower of conditions, which exists if the type theory has Reedy limits of diagrams over the ordinal omega. If B is an ntype for some given finite n, (nontrivial) Reedy limits are unnecessary, allowing us to construct functions from the truncation of A to B in homotopy type theory without further assumptions.
To obtain these results, we develop some theory on equality diagrams, especially equality semisimplicial types. In particular, we show that the semisimplicial equality type over any type satisfies the Kan condition, which can be seen as the simplicial version of the fundamental result by Lumsdaine, and by van den Berg and Garner, that types are weak omegagroupoids.
Finally, we present some results related to formalisations of infinite structures that seem to be impossible to express internally. To give an example, we show how the simplex category can be implemented so that the categorical laws hold strictly. In the presence of very dependent types, we speculate that this makes the Reedy approach for the famous open problem of defining semisimplicial types work.
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