TypeTopology Various new theorems in univalent mathematics written in Agda ------------------------------------------------------------- Martin Escardo and collaborators, 2010--2024--∞, continuously evolving. https://www.cs.bham.ac.uk/~mhe/ https://github.com/martinescardo/TypeTopology Tested with Agda 2.6.4.3 * Our main use of this development is as a personal blackboard or notepad for our research and that of collaborators. In particular, some modules have better and better results or approaches, as time progresses, with the significant steps kept, and with failed ideas and calculations eventually erased. * We offer this page as a preliminary announcement of results to be submitted for publication, of the kind we would get when we visit a mathematician's office. * We have also used this development for learning other people's results, and so some previously known constructions and theorems are included (sometimes with embellishments). * The required material on HoTT/UF has been developed on demand over the years to fulfill the needs of the above as they arise, and hence is somewhat chaotic. It will continue to expand as the need arises. Its form is the result of evolution rather than intelligent design (paraphrasing Linus Torvalds). Our lecture notes develop HoTT/UF in Agda in a more principled way, and offers better approaches to some constructions and simpler proofs of some (previously) difficult theorems. (https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/) Our philosophy, here and in the lecture notes, is to work with a minimal Martin-Löf type theory, and use principles from HoTT/UF (existence of propositional truncations, function extensionality, propositional extensionality, univalence, propositional resizing) and classical mathematics (excluded middle, choice, LPO, WLPO) as explicit assumptions for the theorems, or for the modules, that require them. As a consequence, we are able to tell very precisely which assumptions of HoTT/UF and classical mathematics, if any, we have used for each construction, theorem or set of results. We also work, deliberately, with a minimal subset of Agda. See below for more about the philosophy. * There is also a module that links some "unsafe" modules that use type theory beyond MLTT and HoTT/UF, which cannot be included in this safe-modules index: The system with type-in-type is inconsistent (as is well known), countable Tychonoff, and compactness of the Cantor type using countable Tychonoff. (https://www.cs.bham.ac.uk/~mhe/TypeTopology/AllModulesIndex.html) * In our last count, on 22nd February 2024, this development has 722 files and 204k lines of code, including comments and blank lines. But we don't update the count frequently. Philosophy of the repository ---------------------------- * We adopt the univalent point of view, even in modules which don't assume the univalence axiom. In particular, we take seriously the distinction between types that are singletons (contractible), propositions, sets, 1-groupoids etc., even when the univalence axiom, or its typical consequences such as function extensionality and propositional extensionality, are not needed to reason about them. * We work in a minimal version of intensional Martin Löf Type Theory, with very few exceptions, which we refer to as Spartan MLTT. This is compatible with the UniMath approach. * We adopt the Agda flag exact-split, so that Agda definitions by pattern matching are definitional equalities, to stay as close as Agda can check to the above MLTT. * We work in a minimal subset of Agda to implement Spartan MLTT and work with it. In particular, we restrict ourselves to safe features (with the flags --safe --no-sized-types --no-guardedness). * Some functions, and theorems, and definitions need HoTT/UF axioms. They are always given explicitly as assumptions. Postulates are not allowed in this development. * The development is mostly constructive. A few theorems have non-constructive, explicit assumptions, such as excluded middle, or choice and global choice. One example is Cantor-Schröder-Bernstein for arbitrary (homotopy) types, which was published in the Journal of Homotopy and Related Structures (written in mathematical vernacular as advanced in the HoTT book and originally proposed by Peter Aczel). * We don't assume propositional resizing as Voevodsky and UniMath do. But there are some theorems whose hypotheses or conclusions involve propositional resizing (as an axiom, rather than as a rule of the type theory as unimath does). * The general idea is that any theorem here should be valid in any ∞-topos. * In particular, we don't use Cubical Agda features, deliberately, because at present it is not known whether (some) cubical type theory has an interpretation in any ∞ topos. * However, by fulfilling the HoTT hypotheses with Cubical-Agda implementations, we should be able to run the constructions and proofs given here, so that we get constructivity in the computational sense (as opposed to constructivity in the sense of validity in any (∞-)topos). Click at the imported module names to navigate to them: \begin{code} {-# OPTIONS --safe --without-K #-} module index where import BinarySystems.index import CantorSchroederBernstein.index import Categories.index import Circle.index import CoNaturals.index import ContinuityAxiom.index import Coslice.index import CrossedModules.index import DedekindReals.index import DomainTheory.index import Dominance.index import Duploids.index import Dyadics.index import DyadicsInductive.index import EffectfulForcing.index import Factorial.index import Field.index import Fin.index import Games.index import Groups.index import InjectiveTypes.index import Integers.index import Iterative.index import Lifting.index import Locales.index import MGS.index import MLTT.index import MetricSpaces.index import Modal.index import Naturals.index import Notation.index import NotionsOfDecidability.index import OrderedTypes.index import Ordinals.index import PCF.index import PathSequences.index import Quotient.index import Rationals.index import Relations.index import Slice.index import TWA.index import Taboos.index import TypeTopology.index import UF.index import Various.index import W.index import WildCategories.index import gist.index \end{code} TODO. Explain what each of the above does here. The above includes only the --safe modules. A list of all modules is here: https://www.cs.bham.ac.uk/~mhe/TypeTopology/AllModulesIndex.html