groupoid: meaning, definition, pronunciation and examples
Very LowTechnical / Academic
Quick answer
What does “groupoid” mean?
A set equipped with a single binary operation that is associative and has an identity element, but where not every element necessarily has an inverse.
Audio
Pronunciation
Definition
Meaning and Definition
A set equipped with a single binary operation that is associative and has an identity element, but where not every element necessarily has an inverse.
In category theory, a small category in which every morphism is an isomorphism. In a broader, informal sense, it can refer to any algebraic structure resembling a group but with relaxed axioms.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling conventions follow the standard UK/US pattern for the suffix '-oid' (no variation).
Connotations
None beyond its strict mathematical definition.
Frequency
Extremely rare in both varieties, confined to highly specialised academic discourse.
Grammar
How to Use “groupoid” in a Sentence
The [fundamental] groupoid of [a topological space]A groupoid [is defined as] [a set with a partial operation]Vocabulary
Collocations
Examples
Examples of “groupoid” in a Sentence
verb
British English
- The construction groupoids the path components of the space.
- One can groupoid the set of equivalences.
American English
- The construction groupoids the path components of the space.
- One can groupoid the set of equivalences.
adverb
British English
- The elements act groupoidally on the set.
- The operation is defined groupoid-wise.
American English
- The elements act groupoidally on the set.
- The operation is defined groupoid-wise.
adjective
British English
- The groupoid structure is essential for the proof.
- We consider the groupoid category.
American English
- The groupoid structure is essential for the proof.
- We consider the groupoid category.
Usage
Meaning in Context
Business
Not used.
Academic
Primary context. Used in advanced mathematics papers, textbooks, and seminars on algebra, topology, or category theory.
Everyday
Virtually never used.
Technical
Secondary context. May appear in theoretical computer science, especially concerning semantics or algebraic specification.
Vocabulary
Synonyms of “groupoid”
Strong
Neutral
Weak
Vocabulary
Antonyms of “groupoid”
Watch out
Common Mistakes When Using “groupoid”
- Using 'groupoid' to mean a small group (e.g., a team or clique).
- Pronouncing it as /ɡrʊpɔɪd/ (like 'group' with a short vowel).
- Assuming it is a common or general term outside mathematics.
FAQ
Frequently Asked Questions
No. A group is a special case of a groupoid where the binary operation is defined for all pairs of elements and every element has an inverse. A groupoid is a more general structure.
Consider a set of journeys between cities, where a journey from A to B can be combined with a journey from B to C to get a journey from A to C. Not all combinations are possible (you can't combine a journey from A to B with one from C to D), and every journey can be reversed. This structure forms a groupoid.
Groupoids are useful for describing local or partial symmetries and structures that are not globally uniform. For example, the fundamental groupoid of a topological space can describe loops based at different points and how they relate, whereas the fundamental group only describes loops based at a single chosen point.
Almost never. It is a highly specialised technical term with no significant usage in everyday language, business, or other academic fields like humanities or social sciences.
Groupoid is usually technical / academic in register.
Groupoid: in British English it is pronounced /ˈɡruːpɔɪd/, and in American English it is pronounced /ˈɡrupɔɪd/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None”
Learning
Memory Aids
Mnemonic
Think: 'GROUP-like but OID (resembling)'. It resembles a group but doesn't have all the full group properties.
Conceptual Metaphor
A SOCIAL NETWORK WITH RESTRICTED INTERACTIONS: Like a group where everyone can interact, a groupoid is like a network where connections (operations) are only possible between certain specific pairs of people (elements).
Practice
Quiz
In which field is the term 'groupoid' primarily used?