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
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
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
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
$\mathbb{R}$-cohesive homotopy type theory or real-cohesive homotopy type theory is a version of cohesive homotopy type theory which has the axiom of real cohesion. It provides a synthetic foundation for topology, real analysis, classical homotopy theory, and algebraic topology.
We assume a spatial type theory presented with crisp term judgments $a::A$. In addition, we also assume the spatial type theory has a Dedekind real numbers type $\mathbb{R}$, and $\mathbb{R}$-localizations $\mathcal{L}_{\mathbb{R}}(-)$.
Given a type $A$, let us define $\mathrm{const}_{A, \mathbb{R}}:A \to (\mathbb{R} \to A)$ to be the type of all constant functions in the Dedekind real numbers $\mathbb{R}$:
There is an equivalence $\mathrm{const}_{A, 1}:A \simeq (1 \to A)$ between the type $A$ and the type of functions from the unit type $1$ to $A$. Given types $B$ and $C$ and a function $F:(B \to A) \to (C \to A)$, type $A$ is $F$-local if the function $F:(B \to A) \to (C \to A)$ is an equivalence of types.
A crisp type $\Xi \vert () \vdash A$ is discrete if the function $(-)_\flat:\flat A \to A$ is an equivalence of types.
The axiom of $\mathbb{R}$-cohesion states that for the crisp affine line $\Xi \vert () \vdash \mathbb{R} \; \mathrm{type}$, given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{R}})$-local.
This allows us to define discreteness for non-crisp types: a type $A$ is discrete if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{R}})$-local.
The shape modality in $\mathbb{R}$-cohesive homotopy type theory is then defined as the $\mathbb{R}$-localization $\esh(A) \coloneqq \mathcal{L}_{\mathbb{R}}(A)$, which ensures that the shape of $\mathbb{R}$ itself is a contractible type.
Last revised on November 15, 2022 at 21:26:22. See the history of this page for a list of all contributions to it.