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Abstract
In this paper, we explore, enrich, and otherwise mildly generalise a prominent definition of weak ncategory by Batanin, as refined by Leinster, to give a definition of weak ndimensional Vcategory, 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
wellbehaved. Our leading example of such a V is the category ωCpo, ωCpoenriched 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 Vgraphs and small Vcategories. Iterating, this produces categories of ndimensional Vgraphs and strict ndimensional Vcategories that
are also locally presentable and extensive. We show that the free strict ndimensional Vcategory monad on the category of ndimensional Vgraphs is cartesian. This, along with results due to Garner, allows us to follow Batanin and Leinster’s approach for defining weak ncategories. In the case that V = Set, the resulting definition of weak ndimensional Vcategory agrees with
Leinster’s definition.
wellbehaved. Our leading example of such a V is the category ωCpo, ωCpoenriched 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 Vgraphs and small Vcategories. Iterating, this produces categories of ndimensional Vgraphs and strict ndimensional Vcategories that
are also locally presentable and extensive. We show that the free strict ndimensional Vcategory monad on the category of ndimensional Vgraphs is cartesian. This, along with results due to Garner, allows us to follow Batanin and Leinster’s approach for defining weak ncategories. In the case that V = Set, the resulting definition of weak ndimensional Vcategory agrees with
Leinster’s definition.
Original language  English 

Publication status  Acceptance date  2018 
Keywords
 higher dimensional category
 enriched category
 extensive category
 cartesian monad
ASJC Scopus subject areas
 General Computer Science
 General Mathematics
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 1 Finished

Higher Category Theoretic Structure of Programming Semantics
Power, J. (PI)
11/11/16 → 11/11/18
Project: Research council