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
In cohesive type theory, the axiom of sufficient cohesion states that there is a type with elements and such that and the shape of is a contractible type.
This is equivalent in strength to axiom C2, which says that there is a type with elements and such that and every crisp type is discrete if and only if every function from into is a constant function. is a modality which takes types in the crisp mode to its corresponding discrete type in the cohesive mode, and a discrete type in the cohesive mode is one for which the canonical function is an equivalence of types.
Shulman 2018 showed that axiom C2 implies axiom C0, which implies that every function from into a discrete type is a constant function; conversely, if every function from to a discrete type is constant, then it holds for the discrete types which are in the image of the modality. Finally, Aberlé 2024 proved that the axiom of sufficient cohesion holds if and only if every function from into a discrete type is a constant function.
The shape modality of a type is defined to be the localization of at the type
Assuming the axiom of sufficient cohesion, there exists a pullback which is not preserved by the shape modality.
By the recursion principle of the positive unit type, there are functions and . By the fact that , the pullback of these two functions is the empty type . However, while the shape of is always contractible and the shape of is contractible by the axiom of sufficient cohesion, the shape of is still , which is not contractible.
William Lawvere. Axiomatic cohesion. Theory and Applications of Categories, 19(3):41–49, 2007.
Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)
C.B. Aberlé, Parametricity via Cohesion [arXiv:2404.03825]
Last revised on August 21, 2024 at 16:20:37. See the history of this page for a list of all contributions to it.