under construction
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Backround
Definition
Presentation over a site
Models
By cohesive homotopy type theory one will mean a modal homotopy type theory implementing cohesive homotopy theory via an adjoint triple of modal operators (shape modality flat modality sharp modality), hence with categorical semantics in cohesive -toposes.
A first formulation of cohesive homotopy type theory [Schreiber & Shulman (2012)] added the required adjoint triple of modal operators as axioms to plain homotopy type theory.
Another approach [Shulman (2015)] is to change the underlying rules of dependent type theory itself by adjoining syntax for flat-modal contexts (“crisp contexts”).
This second approach, via a modified type theory with crisp contexts (which has meanwhile be implemented in actual proof assistants such as Agda-flat), better lends itself to producing proofs internal to the theory and is what most authors now mean by (real-)cohesive homotopy type theory, see e.g. developments in: Myers (2019), Myers (2021), Myers & Riley (2023).
In any case, Cohesive homotopy type theory is an axiomatic theory of the higher geometry of cohesive homotopy theory, i.e. of the homotopy theory of differential topology:
In its categorical semantics, the types in cohesive HoTT are interpreted as cohesive homotopy types, hence as cohesive ∞-groupoids, such as for instance smooth ∞-groupoids. See also at motivation for cohesive toposes for a non-technical discussion.
We discuss the formulation in homotopy type theory of the internal axioms on a cohesive (∞,1)-topos.
Cohesive homotopy type theory is a modal type theory which adds to homotopy type theory an adjoint triple of modalities
called
where and are idempotent monads and where is an idempotent comonad, subject to some compatibility condition.
Axiom A. The ambient homotopy type theory has a left-exact reflective sub-(∞,1)-category, to be called the base (∞,1)-topos “of codiscrete objects”.
Coq code at Codiscrete.v
We write
for the reflector into codiscrete objects.
The homotopy type theory of the codiscrete objects we call the external theory.
Axiom B. There is also a coreflective sub-(∞,1)-category of discrete objects such that with the codiscrete reflection it makes the ambient theory that of a local (∞,1)-topos.
Coq code at LocalTopos.v.
The coreflector from discrete objects we write
Axiom C The discrete objects are also reflective, the reflector is left adjoint to the coreflector and preserves product types.
Coq code at CohesiveTopos.v.
We write
for the reflector into discrete objects.
There is another way to define cohesive homotopy type theory, as a multimodal type theory. Multimodal type theories were first introduced in Gratzer, Kavvos, Nuyts, & Birkedal 2021 and consists of a strict 2-category called the mode theory, whose objects are called modes and whose 1-cells are called modalities. For each mode , there is a type theory at mode , and for each 1-cell between objects there is a type theoretic modality in the type theory, which comes with its associated context lock.
The mode theory of cohesive homotopy type theory consists of
two modes, the crisp or discrete mode and the cohesive mode ,
1-cells and ,
2-cells
which satisfy the following triangle identities:
which makes the 1-cells into an adjoint quadruple
The notations for the adjoint quadruple are derived from the introduction of Shulman 2015, but are traditionally expressed as
(i.e. see cohesive infinity-topos). However, we do not use the above notations because conflicts with the use of for the dependent product type (i.e. ) and conflicts with the use of to express arbitrary contexts in inference rules in dependent type theory.
The sharp, flat, and shape endo-modalities on the cohesive mode can be defined as composites of the modalities above:
yielding the adjoint triple
In the multimodal type theory associated with cohesive homotopy type theory, there are two type judgments:
which says that is a type in the crisp mode;
which says that is a type in the cohesive mode.
Similarly, there are two term judgments:
which says that is a term of type in the crisp mode;
which says that is a term of type in the cohesive mode.
and two context judgments:
which says that is a context in the crisp mode;
which says that is a context in the cohesive mode.
as well as two separate judgments each for judgmental equality of types and terms:
which says that and are judgmentally equal types in the crisp mode;
which says that and are judgmentally equal types in the cohesive mode;
which says that and are judgmentally equal terms of type in the crisp mode;
which says that and are judgmentally equal terms of type in the cohesive mode.
which says that and are judgmentally equal contexts in the crisp mode;
which says that and are judgmentally equal contexts in the cohesive mode;
The original papers on multimodal type theory use the symbol @ instead of but it doesn’t seem to be possible to put @ inside of latex math mode on the nLab, whether directly or inside the mathrm command.
Then we have the rules for the context locks of the modalities of cohesive homotopy type theory:
The original papers also used a lock symbol that the author does not know how to replicate on the nLab.
(…)
(Kolomatskaia & Shulman 2023 might also be useful here in defining the type theory)
Before looking at the consequences of the axioms formally, we mention some example phenomena to illustrate the meaning of the axioms.
We indicate one central aspect of geometric homotopy theory that is not visible in plain homotopy type theory, but is captured by its cohesive refinement.
The standard interval in topological spaces plays two rather different roles, depending on what kind of equivalence between spaces is considered. To make this more vivid, it serves to think of as equipped even with its canonical structure of a smooth manifold (with boundary).
The canonical map to the point is certainly not a diffeomorphism, and from the point of view of differential geometry the interval carries non-trivial structure. Notably its endpoints are not equivalent points (terms) in differential geometry, but are distinct. From the point of view of differential geometry the interval is a homotopy 0-type (has h-level 2) – but one that is in some way equipped with geometric structure.
This geometric structure, however, induces also a notion of geometric paths in the interval, such that any two of its points are connected by such a path, after all. In other words, one can form the smooth fundamental ∞-groupoid of the interval and regard that as a homotopy type without further geometric structure (a discrete ∞-groupoid). This is an interval type, while itself is not.
As such, the canonical map is an equivalence after all, namely a weak homotopy equivalence. Therefore, after application of , what used to be a geometric 0-type becomes a (-1)-type and actually a (-2)-type (h-level 0) – up to equivalence the interval type, but without any geometry.
This latter property is what makes the interval important in bare homotopy theory, where it serves to model notions such as cylinder objects, left homotopies, etc. The former property, however, is what makes the interval important in geometry, where it serves to model Cartesian spaces, manifolds, etc.
In cohesive homotopy type theory these two roles of the interval can both be seen, via the reflective embedding of discrete objects, and the transition between them is present, via the fundamental ∞-groupoid reflector .
Specifically, there is a model for homotopy cohesion, called Smooth∞Grpd, in which smooth manifolds (with boundary) are fully faithfully embedded, where hence exists as a type that behaves as the interval in differential geometry, and where is equivalent to the unit type.
More generally, in this model every smooth manifold is a homotopy 0-type/0-truncated object, but the type is a discrete ∞-groupoid whose homotopy type is that of the topological space underlying , as regarded in the standard homotopy category of topological spaces.
In particular, the smooth circle in this model is a 0-type such that is the 1-type (the delooping groupoid of the integers).
One can turn this around and axiomatize a continuum line object in cohesive homotopy type theory as a ring object such that .
We discuss implications of the axioms of cohesive homotopy type theory and go through the discussion of the various structures in a cohesive (∞,1)-topos.
For and two types, the externalization of the function type is the type of cocycles on with coefficients in . Its h-level 2 truncation is the cohomology of with coefficients in .
We give the Coq-formalization of Flat cohomology and local systems.
For a type, we say that cohomology with coefficients in is flat cohomology. A cocycle term is called a local system of coefficients on .
(…)
We give the Coq-formalization of intrinsic de Rham cohomology.
The homotopy fiber type of the coreflection we call the de Rham coefficient type of , denoted . So there is a fiber sequence
Coq-code:
Require Import Homotopy Subtopos Codiscrete LocalTopos CohesiveTopos.
Hypothesis BG : Type.
Hypothesis BG_is_0connected : is_contr (pi0 BG).
Hypothesis pt : BG.
Definition flat_dR : #Type
:= ipullback ([[fun _:unit => pt]]) (from_flat ([BG])).
We give the Coq-formalization of Differential cohomology.
(…)
First discussion of a (homotopy) type theoretic formulation of the modal cohesive homotopy theory (adopting terminology from Lawvere 2007 ) considered in
was given in
following
by adding axioms for the adjoint triple of modal operators to plain homotopy type theory.
See also broader discussion in:
Urs Schreiber, Modern Physics formalized in Modal Homotopy Type Theory (2016)
David Corfield, Chap. 5 of Modal Homotopy Type Theory, Oxford University Press (2020) [ISBN:9780198853404]
Another type-theoretic formulation of cohesive homotopy theory, now obtained by changing the rewrite rules of type theory itself – adding a syntactic notion of flat-modal (“crisp”) contexts:
following a general pattern for modal type theory laid out in
with exposition in:
This approach (also “real cohesive type theory”) is now what most people refer to when speaking of cohesive homotopy type theory.
Notice that at this point there is no proof assistant that actually implements the shape modality this way, only the system consisting of flat modality sharp modality (spatial type theory) runs on computers: eg. via Agda-flat.
Discussion of a fragment of differential cohesive homotopy type theory with Agda-flat:
Further development of (real-)cohesive homotopy type theory:
Formalization of the shape/flat-fracture square (differential cohomology hexagon):
David Jaz Myers, Modal Fracture of Higher Groups [arXiv:2106.15390]
also: talk at CMU-HoTT Seminar, 2021 (pdf, pdf)
Discussion of pairs of commuting cohesive structures (such as the combination of real cohesion and equivariant relevant for differential orbifold cohomology:
Exposition in:
For parametricity in cohesive homotopy type theory:
The development of cohesive homotopy type theory as a multimodal type theory uses material from
Daniel Gratzer, G. Alex Kavvos, Andreas Nuyts, Lars Birkedal: Multimodal Dependent Type Theory, Logical Methods in Computer Science 17 3 (2021) lmcs:7713 [arXiv:2011.15021, doi:10.46298/lmcs-17(3:11)2021]
Astra Kolomatskaia, Michael Shulman, Displayed Type Theory and Semi-Simplicial Types [arXiv:2311.18781]
Last revised on August 21, 2024 at 11:11:51. See the history of this page for a list of all contributions to it.