TY - JOUR

T1 - Flat vs. filtered colimits in the enriched context

AU - Lack, Stephen

AU - Tendas, Giacomo

PY - 2022/8/6

Y1 - 2022/8/6

N2 - The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as Ab,SSet,Cat and Met. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: 1. give sufficient conditions on V so that (a) ⇔ (b) holds; 2. give sufficient conditions on V so that (a) ⇔ (b) holds up to Cauchy completion; 3. explore some examples not covered by (1) or (2).

AB - The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as Ab,SSet,Cat and Met. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: 1. give sufficient conditions on V so that (a) ⇔ (b) holds; 2. give sufficient conditions on V so that (a) ⇔ (b) holds up to Cauchy completion; 3. explore some examples not covered by (1) or (2).

UR - https://researchers.mq.edu.au/en/publications/8b1e243e-a411-42f2-b7eb-4593fd77985b

U2 - 10.1016/j.aim.2022.108381

DO - 10.1016/j.aim.2022.108381

M3 - Article

SN - 0001-8708

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -