2023/10/06 by Jason Parker, Parker, Jason
Mathematics · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Logic in Computer Science (cs.LO)
paper · pdf · doi:10.48550/arxiv.2310.04587
openalex publication_date 2023/10/06 · openalex created_date 2023/10/13 · openalex updated_date 2026/07/28
It is a classical result of categorical algebra, due to Lawvere and Linton, that finitary varieties of algebras (in the sense of Birkhoff) are dually equivalent to finitary monads on Set. Recent work of Adámek, Dostál, and Velebil has established that analogous results also hold in certain enriched contexts. Specifically, taking V to be one of the cartesian closed categories Pos, UltMet, ω-CPO, or DCPO of respectively posets, (extended) ultrametric spaces, ω-cpos, or dcpos, Adámek, Dostál, and Velebil have shown that a suitable category of V-enriched varieties of algebras is dually equivalent to the category of strongly finitary V-monads on V. In this paper, we extend and generalize these results in two ways: by allowing V to be an arbitrary complete and cocomplete cartesian closed category that is concrete over Set, and by also considering the multi-sorted case. Given a set S of sorts, we define a suitable notion of (finitary) V-enriched S-sorted variety, and we say that a V-monad on the product V-category VS is strongly finitary if its underlying V-endofunctor is the left Kan extension of its restriction to a suitable full sub-V-category of VS. Our main result is that the category of V-enriched S-sorted varieties is dually equivalent to the category of strongly finitary V-monads on VS. By taking S to be a singleton and V to be Pos, UltMet, ω-CPO, or DCPO, we thus recover the aforementioned results of Adámek, Dostál, and Velebil. We provide several classes of examples of V-enriched S-sorted varieties, many of which admit very concrete, syntactic formulations.