Monads and theories

Abstract

Given a locally presentable enriched category E together with a small dense full subcategory A of arities, we study the relationship between monads on E and identity-on-objects functors out of A, which we call Apretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised as the A-nervous monads-those for which the conclusions of Weber's nerve theorem hold-and the A-theories, which we introduce here.The resulting equivalence between A-nervous monads and A-theories is best possible in a precise sense, and extends almost all previously known monad-theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak ω-groupoids.Besides establishing our general correspondence and illustrating its reach, we study good properties of A-nervous monads and A-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Melliès and Weber. arXiv:1805.04346v1 [math.CT] 11 May 2018 K G G J b b D Explicitly, J is given on objects by Ja = Ha, and on homs by B(a, b)

Metric monads

“…Definition 3.1. (Bourke and Garner (2019)). Let K be a category and A a small dense subcategory of K .…”

“…The resulting enriched Lawvere theories from Nishizawa and Power (2009) do not fit our aim to do universal algebra over a V -category K . Recently, Bourke and Garner (2019) introduced A -pretheories for every small dense subcategory A of K and related them to A -nervous enriched monads on K . Their pretheories perfectly suit our needs and describe λ-ary monads on every locally λ-presentable Vcategory K provided that V is locally λ-presentable as a closed category.…”

“…Their pretheories perfectly suit our needs and describe λ-ary monads on every locally λ-presentable Vcategory K provided that V is locally λ-presentable as a closed category. Indeed, if A is the full subcategory of λ-presentable objects of K the A -pretheory of Bourke and Garner (2019) is given by (X, Y)-ary operations and equations; here, X and Y are λ-presentable objects of K . Over Met, we can apply Di Liberti and Rosický (2009) and relate equational theories whose operations have finite metric spaces as arities to monads preserving directed colimits of isometries.…”

“…For the comfort of (some) readers, we will do the unenriched case at first. In Section 3, we show that λ-ary pretheories of Bourke and Garner (2019) can be presented as equational theories over a category K introduced in Rosický (1981) (motivated by Linton (1969)). We show how it can be used to prove the result from Bourke and Garner (2019) that λ-ary pretheories over a locally λpresentable category K correspond to λ-ary monads on K .…”

“…In Section 3, we show that λ-ary pretheories of Bourke and Garner (2019) can be presented as equational theories over a category K introduced in Rosický (1981) (motivated by Linton (1969)). We show how it can be used to prove the result from Bourke and Garner (2019) that λ-ary pretheories over a locally λpresentable category K correspond to λ-ary monads on K . Then we apply it to the case when K is not locally finitely presentable but only locally finitely generated in the sense of Di Liberti and Rosický (2009), which is the case of metric spaces.…”