homotopy: meaning, definition, pronunciation and examples
Very Low (C2)Highly Technical / Academic (Mathematics)
Quick answer
What does “homotopy” mean?
In topology and abstract algebra, a continuous deformation or transformation between two mathematical objects, particularly between two paths or functions.
Audio
Pronunciation
Definition
Meaning and Definition
In topology and abstract algebra, a continuous deformation or transformation between two mathematical objects, particularly between two paths or functions.
More broadly, it can refer to a concept of equivalence or deformation in formal systems, used metaphorically to describe gradual transformations between abstract entities.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or definitional differences. Spelling and pronunciation are consistent across varieties.
Connotations
None beyond its strict mathematical meaning.
Frequency
Extremely rare outside specialized mathematical discourse in both regions.
Grammar
How to Use “homotopy” in a Sentence
The homotopy between X and YA homotopy from A to BTo show X and Y are homotopicThe space of homotopiesVocabulary
Collocations
Examples
Examples of “homotopy” in a Sentence
verb
British English
- The two maps can be homotoped to a common function.
- We need to homotopy this loop to a point.
American English
- The two functions are homotopic.
- We homotop the initial path to a more convenient one.
adverb
British English
- The spaces are homotopy equivalent.
- The maps agree homotopically.
American English
- The two structures are homotopy related.
- The process works homotopically.
adjective
British English
- The homotopy category is central to the theory.
- They established a homotopy commutative diagram.
American English
- This is a homotopy invariant property.
- They studied the homotopy theoretic model.
Usage
Meaning in Context
Business
Never used.
Academic
Used exclusively in advanced mathematics, theoretical physics, and computer science research.
Everyday
Never used.
Technical
Core term in algebraic topology, category theory, homotopy type theory, and related fields.
Vocabulary
Synonyms of “homotopy”
Strong
Neutral
Weak
Vocabulary
Antonyms of “homotopy”
Watch out
Common Mistakes When Using “homotopy”
- Using it as a general synonym for 'similarity' or 'analogy'.
- Confusing 'homotopy' with 'homeomorphism' (the latter is a specific type of equivalence without tearing or gluing, but is not a continuous *deformation*).
FAQ
Frequently Asked Questions
No. It is a highly specialized technical term used almost exclusively in advanced mathematics and related theoretical fields.
A homeomorphism is a specific one-to-one correspondence between spaces that is continuous in both directions. A homotopy is a broader concept—a continuous *process* of deforming one function or path into another. All homeomorphisms induce a homotopy equivalence, but not all homotopy equivalences are homeomorphisms.
Yes, informally within mathematical discourse. For example, 'We can homotop f to g' means we can demonstrate a continuous deformation (a homotopy) from function f to function g.
It provides a way to classify spaces and maps by their 'shape' in a flexible, algebraic manner. Properties that remain unchanged under homotopy (homotopy invariants) are fundamental tools for distinguishing between different topological spaces.
In topology and abstract algebra, a continuous deformation or transformation between two mathematical objects, particularly between two paths or functions.
Homotopy is usually highly technical / academic (mathematics) in register.
Homotopy: in British English it is pronounced /həʊˈmɒtəpi/, and in American English it is pronounced /hoʊˈmɑːtəpi/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “Path-connected by a homotopy”
Learning
Memory Aids
Mnemonic
Think of 'homo-' (same) + 'topos' (place in Greek, related to topology). A homotopy lets you reshape one thing into another in the same topological 'place' or space without cutting or gluing.
Conceptual Metaphor
A HOMOTOPY IS A CONTINUOUS PATH OF TRANSFORMATIONS (like a movie where one shape smoothly morphs into another).
Practice
Quiz
In which field is the term 'homotopy' primarily used?