dorsal/arxiv
View SchemaHigher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes
| Authors | John C. Baez, James Dolan |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9702014 |
| URL | https://arxiv.org/abs/q-alg/9702014 |
| Journal | Adv. Math. 135 (1998), 145-206. |
Abstract
We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional opetopes'. Opetopes form a category, and presheaves on this category are called `opetopic sets'. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weak omega-categories. Similarly, starting from an arbitrary operad O instead of I, we define `n-coherent O-algebras', which are n times categorified analogs of algebras of O. Examples include `monoidal n-categories', `stable n-categories', `virtual n-functors' and `representable n-prestacks'. We also describe how n-coherent O-algebra objects may be defined in any (n+1)-coherent O-algebra.
{
"annotation_id": "76393e14-0647-4c1b-b92a-6e03125738bc",
"date_created": "2026-03-02T18:01:28.356000Z",
"date_modified": "2026-03-02T18:01:28.356000Z",
"file_hash": "dda154fc4b025bc7cd485b12ba7bdff764c45dc9598de6c7f251adff04ab8731",
"private": false,
"record": {
"abstract": "We give a definition of weak n-categories based on the theory of operads. We\nwork with operads having an arbitrary set S of types, or `S-operads\u0027, and given\nsuch an operad O, we denote its set of operations by elt(O). Then for any\nS-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O.\nLetting I be the initial operad with a one-element set of types, and defining\nI(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional\nopetopes\u0027. Opetopes form a category, and presheaves on this category are called\n`opetopic sets\u0027. A weak n-category is defined as an opetopic set with certain\nproperties, in a manner reminiscent of Street\u0027s simplicial approach to weak\nomega-categories. Similarly, starting from an arbitrary operad O instead of I,\nwe define `n-coherent O-algebras\u0027, which are n times categorified analogs of\nalgebras of O. Examples include `monoidal n-categories\u0027, `stable n-categories\u0027,\n`virtual n-functors\u0027 and `representable n-prestacks\u0027. We also describe how\nn-coherent O-algebra objects may be defined in any (n+1)-coherent O-algebra.",
"arxiv_id": "q-alg/9702014",
"authors": [
"John C. Baez",
"James Dolan"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Adv. Math. 135 (1998), 145-206.",
"title": "Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes",
"url": "https://arxiv.org/abs/q-alg/9702014"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "dbd9e791-6a17-4afd-bbc2-2177bfcf8926",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}