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Homotopical algebraic geometry: The main idea is to extend schemes by formally replacing the rings with any kind of "homotopy-ring-like object". More precisely this object is a commutative monoid in a symmetric monoidal category endowed with a notion of equivalences which are understood as "up-to-homotopy monoid" (e.g. E ∞-rings). 2002: Peter ...
The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a ...
A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves (or timelike multiply connected ).
The Thom isomorphism brings cobordism of manifolds into the ambit of homotopy theory. 1952: Edwin E. Moise: Moise's theorem established that a 3-dimension compact connected topological manifold is a PL manifold (earlier terminology "combinatorial manifold"), having a unique PL structure. In particular it is triangulable. [37]
In general, every manifold has the homotopy type of a CW complex; [3] in fact, Morse theory implies that a compact manifold has the homotopy type of a finite CW complex. [citation needed] Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout. It satisfies the universal property of a pullback up to homotopy. [ citation needed ] Concretely, given f : X → Z {\displaystyle f:X\to Z} and g : Y → Z {\displaystyle g:Y\to Z} , it can be constructed as
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
This is a timeline of bordism, a topological theory based on the concept of the boundary of a manifold. For context see timeline of manifolds . Jean Dieudonné wrote that cobordism returns to the attempt in 1895 to define homology theory using only (smooth) manifolds.