(topology) A continuous deformation of one continuous function or map to another.
Used in a Sentence
homotopy
Definitions, parts of speech, synonyms, and sentence examples for homotopy.
Editorial note
I don't think synthetic homotopy type theory is known to cover all of conventional homotopy theory, so it's work in progress.
Examples19
Definitions4
Parts of speech1
Quick take
(topology) A continuous deformation of one continuous function or map to another.
Meaning at a glance
The clearest senses and uses of homotopy gathered in one view.
(uncountable) The relationship between two continuous functions where homotopy from one to the other is evident.
(topology) A theory associating a system of groups with each topological space.
Definitions
Core meanings and parts of speech for homotopy.
noun
(topology) A continuous deformation of one continuous function or map to another.
noun
(uncountable) The relationship between two continuous functions where homotopy from one to the other is evident.
noun
(topology) A theory associating a system of groups with each topological space.
noun
(topology) A system of groups associated with a topological space.
Example sentences
I don't think synthetic homotopy type theory is known to cover all of conventional homotopy theory, so it's work in progress.
Can someone give me an absolute layman explanation to homotopy type theory (and why it's so interesting)?
This is the same Voevodsky that came up with the univalence axiom[1] that is foundational in HoTT[2] (homotopy type theory)?
Once you add the topology section to the Curry-Howard Isomorphism via homotopy type theory it all starts to merge anyway!
Apparently, homotopy theorists had for a long time been aware that instead of basing mathematics on sets, and then constructing equivalence classes on top of those, you could take homotopy types as primitive..
Today, their ideas are seeing a resurgence in the homotopy type theorists (HTTs).
Another extension that homotopy theory theory does is with user defined types.
I'd take a look at denotational semantics and Homotopy type theory for some existing formalisms.
I knew about (not that I understand it) the connection homotopy...mostly just worried there's work happening that isn't informed by research.
Another achievement is formulation of synthetic homotopy type theory, i.e.
We have something between the two in HoTT - universes of types is stratified by homotopy levels, corresponding to how many dimensions of structure a type has.
First the type theory part of homotopy type theory.
Quote examples
In HoTT, one can define (and reason about) the basic concepts of homotopy theory straight from "day one".
Secondly, HoTT is the first foundation with which computer-formalized proofs of nontrivial theorems in a subject called "homotopy theory" are possible.
Homotopy type theory formalizes informal reasoning style when things which have "the same structure" are considered equal in a constructive type theory setting.
This discussion goes way beyond an "absolute layman explanation to homotopy type theory"...
Proper noun examples
Homotopy Type Theory is a total language, so even type-level programming simply has to terminate, or the proof-checker won't accept it.
Homotopy type theory: extends type theory with proofs of equality in a proper way.
Homotopy type theory fixes equality in type theory.
Frequently asked questions
Short answers drawn from the clearest meanings and examples for this word.
How do you use homotopy in a sentence?
I don't think synthetic homotopy type theory is known to cover all of conventional homotopy theory, so it's work in progress.
What does homotopy mean?
(topology) A continuous deformation of one continuous function or map to another.
What part of speech is homotopy?
homotopy is commonly used as noun.