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
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
On a local topos/local (∞,1)-topos $\mathbf{H}$, hence equipped with a fully faithful extra right adjoint $coDisc$ to the global section geometric morphism $(Disc \dashv \Gamma)$, is induced an idempotent monad $\sharp \coloneqq coDisc \circ \Gamma$, a modality which we call the sharp modality. This is itself the right adjoint in an adjoint modality with the flat modality $\flat \coloneqq Disc \circ \Gamma$.
We assume a dependent type theory with crisp term judgments $a::A$ in addition to the usual (cohesive) type and term judgments $A \; \mathrm{type}$ and $a:A$, as well as context judgments $\Xi \vert \Gamma \; \mathrm{ctx}$ where $\Xi$ is a list of crisp term judgments, and $\Gamma$ is a list of cohesive term judgments. A crisp type is a type in the context $\Xi \vert ()$, where $()$ is the empty list of cohesive term judgments. In addition, we also assume the dependent type theory has typal equality and judgmental equality.
From here, there are two different notions of the sharp modality which could be defined in the type theory, the strict sharp modality, which uses judgmental equality in the computation rule and uniqueness rule, and the weak sharp modality, which uses typal equality in the computation rule and uniqueness rule. When applied to a type they result in strict sharp types and weak sharp types respectively. The natural deduction rules for strict and weak sharp types are provided as follows:
Weak sharp modalities are primarily used in cohesive weak type theories, while strict sharp modalities are typically used in cohesive type theories which are not weak, such as cohesive Martin-Löf type theory (cohesive homotopy type theory or cohesive higher observational type theory.
The codiscrete objects in a local topos are the modal types for the sharp modality.
The discrete objects in a local topos are the modal types for the flat modality.
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
The terminology of the sharp-modality in the above sense was introduced – in the language of $(\infty,1)$-toposes and as part of the axioms on “cohesive $(\infty,1)$-toposes” – in:
See also the references at local topos.
Early discussion in view of homotopy type theory and as part of a set of axioms for cohesive homotopy type theory is in
The dedicated type theory formulation with “crisp” types, as part of the formulation of real cohesive homotopy type theory, is due to:
Last revised on November 4, 2023 at 23:13:05. See the history of this page for a list of all contributions to it.