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
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
symmetric monoidal (∞,1)-category of spectra
$\mathbb{A}^1$-cohesive homotopy type theory, affine-cohesive homotopy type theory, or algebraic-cohesive homotopy type theory is a version of cohesive homotopy type theory which has an affine line type $\mathbb{A}^1$ and the axiom of $\mathbb{A}^1$-cohesion. $\mathbb{A}^1$-cohesive homotopy type theory provides a synthetic foundation for $\mathbb{A}^1$-homotopy theory, where the affine line is the standard affine line in the Nisnevich site of smooth schemes of finite type over a Noetherian scheme.
Mitchell Riley‘s bunched linear homotopy type theory [Riley (2022)] is a synthetic foundations for parameterized stable homotopy theory. This means that there should be an $\mathbb{A}^1$-cohesive bunched linear homotopy type theory which behaves as a synthetic foundations for motivic homotopy theory.
Similarly to real-cohesive homotopy type theory, we assume a spatial type theory presented with crisp term judgments $a::A$. In addition, we also assume the spatial type theory has an affine line type $\mathbb{A}^1$, and $\mathbb{A}^1$-localizations $\mathcal{L}_{\mathbb{A}^1}(-)$.
Given a type $A$, let us define $\mathrm{const}_{A, \mathbb{A}^1}:A \to (\mathbb{A}^1 \to A)$ to be the type of all constant functions in the affine line $\mathbb{A}^1$:
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 $\mathbb{A}^1$-cohesion states that for the crisp affine line $\Xi \vert () \vdash \mathbb{A}^1 \; \mathrm{type}$, given any crisp type $\Xi \vert () \vdash A \; \mathrm{type}$, $A$ is discrete if and only if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})$-local.
This allows us to define discreteness for non-crisp types: a type $A$ is discrete if $A$ is $(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})$-local.
The shape modality in $\mathbb{A}^1$-cohesive homotopy type theory is then defined as the $\mathbb{A}^1$-localization $\esh(A) \coloneqq \mathcal{L}_{\mathbb{A}^1}(A)$, which ensures that the shape of $\mathbb{A}^1$ itself is a contractible type.
For the presentation of the underlying spatial type theory used for $\mathbb{A}^1$-cohesive homotopy type theory, see:
Last revised on November 16, 2022 at 04:22:15. See the history of this page for a list of all contributions to it.