### Abstract

In this paper, we explore, enrich, and otherwise mildly generalise a prominent definition of weak n-category by Batanin, as refined by Leinster, to give a definition of weak n-dimensional V-category, with a view to applications in programming semantics. We require V to be locally presentable and to be (infinitarily) extensive, a condition which ensures that coproducts are suitably well-behaved. Our leading example of such a V is the category ω-Cpo, ω-Cpo-enriched bicategories already having been used in denotational semantics. We illuminate the implicit use of recursion in Leinster's definition, generating the higher dimensions by a process of repeated enrichment. The key fact is that if V is a locally presentable and extensive category, then so are the categories of small V-graphs and small V-categories. Iterating, this produces categories of n-dimensional V-graphs and strict n-dimensional V-categories that are also locally presentable and extensive. We show that the free strict n-dimensional V-category monad on the category of n-dimensional V-graphs is cartesian. This, along with results due to Garner, allows us to follow Batanin and Leinster's approach for defining weak n-categories. In the case that V = Set, the resulting definition of weak n-dimensional V-category agrees with Leinster's definition.

Original language | English |
---|---|

Pages (from-to) | 73-90 |

Number of pages | 18 |

Journal | Electronic Notes in Theoretical Computer Science |

Volume | 341 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

### Keywords

- cartesian monad
- enriched category
- extensive category
- higher dimensional category

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Electronic Notes in Theoretical Computer Science*,

*341*, 73-90. https://doi.org/10.1016/j.entcs.2018.11.005

**Higher Dimensional Categories : Induction on Extensivity.** / Cottrell, Thomas; Fujii, Soichiro; Power, John.

Research output: Contribution to journal › Article

*Electronic Notes in Theoretical Computer Science*, vol. 341, pp. 73-90. https://doi.org/10.1016/j.entcs.2018.11.005

}

TY - JOUR

T1 - Higher Dimensional Categories

T2 - Induction on Extensivity

AU - Cottrell, Thomas

AU - Fujii, Soichiro

AU - Power, John

PY - 2018/12/1

Y1 - 2018/12/1

N2 - In this paper, we explore, enrich, and otherwise mildly generalise a prominent definition of weak n-category by Batanin, as refined by Leinster, to give a definition of weak n-dimensional V-category, with a view to applications in programming semantics. We require V to be locally presentable and to be (infinitarily) extensive, a condition which ensures that coproducts are suitably well-behaved. Our leading example of such a V is the category ω-Cpo, ω-Cpo-enriched bicategories already having been used in denotational semantics. We illuminate the implicit use of recursion in Leinster's definition, generating the higher dimensions by a process of repeated enrichment. The key fact is that if V is a locally presentable and extensive category, then so are the categories of small V-graphs and small V-categories. Iterating, this produces categories of n-dimensional V-graphs and strict n-dimensional V-categories that are also locally presentable and extensive. We show that the free strict n-dimensional V-category monad on the category of n-dimensional V-graphs is cartesian. This, along with results due to Garner, allows us to follow Batanin and Leinster's approach for defining weak n-categories. In the case that V = Set, the resulting definition of weak n-dimensional V-category agrees with Leinster's definition.

AB - In this paper, we explore, enrich, and otherwise mildly generalise a prominent definition of weak n-category by Batanin, as refined by Leinster, to give a definition of weak n-dimensional V-category, with a view to applications in programming semantics. We require V to be locally presentable and to be (infinitarily) extensive, a condition which ensures that coproducts are suitably well-behaved. Our leading example of such a V is the category ω-Cpo, ω-Cpo-enriched bicategories already having been used in denotational semantics. We illuminate the implicit use of recursion in Leinster's definition, generating the higher dimensions by a process of repeated enrichment. The key fact is that if V is a locally presentable and extensive category, then so are the categories of small V-graphs and small V-categories. Iterating, this produces categories of n-dimensional V-graphs and strict n-dimensional V-categories that are also locally presentable and extensive. We show that the free strict n-dimensional V-category monad on the category of n-dimensional V-graphs is cartesian. This, along with results due to Garner, allows us to follow Batanin and Leinster's approach for defining weak n-categories. In the case that V = Set, the resulting definition of weak n-dimensional V-category agrees with Leinster's definition.

KW - cartesian monad

KW - enriched category

KW - extensive category

KW - higher dimensional category

UR - http://www.scopus.com/inward/record.url?scp=85058042751&partnerID=8YFLogxK

U2 - 10.1016/j.entcs.2018.11.005

DO - 10.1016/j.entcs.2018.11.005

M3 - Article

VL - 341

SP - 73

EP - 90

JO - Electronic Notes in Theoretical Computer Science

JF - Electronic Notes in Theoretical Computer Science

SN - 1571-0661

ER -