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### 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.

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 |
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Publication status | Acceptance date - 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

## Fingerprint Dive into the research topics of 'Higher dimensional categories: recursion on extensivity'. Together they form a unique fingerprint.

## Projects

- 1 Finished

### Higher Category Theoretic Structure of Programming Semantics

Power, J.

11/11/16 → 11/11/18

Project: Research council

## Cite this

Power, A., Cottrell, T., & Fujii, S. (Accepted/In press).

*Higher dimensional categories: recursion on extensivity*.