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 $\dashv$ flat modality $\dashv$ sharp modality), hence with categorical semantics in cohesive $\infty$-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 $ʃ$ and $\sharp$ are idempotent monads and where $\flat$ 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.
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 $I = [0,1]$ 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 $[0,1]$ as equipped even with its canonical structure of a smooth manifold (with boundary).
The canonical map $I \to *$ 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 $0,1 : I$ 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 $\Pi(I)$ of the interval and regard that as a homotopy type without further geometric structure (a discrete ∞-groupoid). This $\Pi(I)$ is an interval type, while $I$ itself is not.
As such, the canonical map $\Pi(I) \to *$ is an equivalence after all, namely a weak homotopy equivalence. Therefore, after application of $\Pi$, 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 $\mathbf{\Pi}$.
Specifically, there is a model for homotopy cohesion, called Smooth∞Grpd, in which smooth manifolds (with boundary) are fully faithfully embedded, where hence $I = [0,1]$ exists as a type that behaves as the interval in differential geometry, and where $\mathbf{\Pi}(I)$ is equivalent to the unit type.
More generally, in this model every smooth manifold $X$ is a homotopy 0-type/0-truncated object, but the type $\mathbf{\Pi}(X)$ is a discrete ∞-groupoid whose homotopy type is that of the topological space underlying $X$, as regarded in the standard homotopy category of topological spaces.
In particular, the smooth circle $S^1$ in this model is a 0-type such that $\mathbf{\Pi}(S^1)$ is the 1-type $\mathbf{B}\mathbb{Z}$ (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 $\mathbb{Z} \hookrightarrow \mathbb{A}^1$ such that $\mathbf{\Pi}(\mathbb{A}^1) \simeq *$.
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 $X$ and $A$ two types, the externalization $\sharp(X \to A)$ of the function type $X \to A$ is the type of cocycles on $X$ with coefficients in $A$. Its h-level 2 truncation $\tau_0 \sharp(X \to Y)$ is the cohomology of $X$ with coefficients in $A$.
We give the Coq-formalization of Flat cohomology and local systems.
For $A$ a type, we say that cohomology with coefficients in $\flat A$ is flat cohomology. A cocycle term $c : \sharp(X \to \flat A)$ is called a local system of coefficients $A$ on $X$.
(…)
We give the Coq-formalization of intrinsic de Rham cohomology.
Let $A = \mathbf{B}G$ be a connected type.
The homotopy fiber type of the coreflection $\flat \mathbf{B}G \to \mathbf{B}G$ we call the de Rham coefficient type of $G$, denoted $\flat_{dR} \mathbf{B}G$. 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 $\dashv$ 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:
Last revised on June 26, 2024 at 12:50:44. See the history of this page for a list of all contributions to it.