Introduction Basic setting and two standard constructions Main theorems Monads and theories John Bourke (joint work with Richard Garner) Department of Mathematics and Statistics Masaryk University CT2018 John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: 1. Finitary monads T on Set . John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: 1. Finitary monads T on Set . 2. Lawvere theories: identity on objects functors F → T that preserve finite coproducts, where F is a skeleton of finite sets. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: 1. Finitary monads T on Set . 2. Lawvere theories: identity on objects functors F → T that preserve finite coproducts, where F is a skeleton of finite sets. ◮ Equivalent approaches – equivalence of categories Mnd f ( Set ) ≃ Law which commutes with semantics. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: 1. Finitary monads T on Set . 2. Lawvere theories: identity on objects functors F → T that preserve finite coproducts, where F is a skeleton of finite sets. ◮ Equivalent approaches – equivalence of categories Mnd f ( Set ) ≃ Law which commutes with semantics. ◮ Many generalisations of this story – other bases than Set , enrichment, other shapes of operations than finite . . . John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: 1. Finitary monads T on Set . 2. Lawvere theories: identity on objects functors F → T that preserve finite coproducts, where F is a skeleton of finite sets. ◮ Equivalent approaches – equivalence of categories Mnd f ( Set ) ≃ Law which commutes with semantics. ◮ Many generalisations of this story – other bases than Set , enrichment, other shapes of operations than finite . . . ◮ Today - a general class of monad–theory correspondences, that arise naturally. Joint work with Richard Garner – see “Monads and theories”(BG18). John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Introduction ◮ Two categorical approaches to classical universal algebra: 1. Finitary monads T on Set . 2. Lawvere theories: identity on objects functors F → T that preserve finite coproducts, where F is a skeleton of finite sets. ◮ Equivalent approaches – equivalence of categories Mnd f ( Set ) ≃ Law which commutes with semantics. ◮ Many generalisations of this story – other bases than Set , enrichment, other shapes of operations than finite . . . ◮ Today - a general class of monad–theory correspondences, that arise naturally. Joint work with Richard Garner – see “Monads and theories”(BG18). ◮ Closely related to, and inspired by, the notions of monad and theories with arities of Berger, Mellies and Weber (BMW12) – but has advantages. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems The basic context ◮ V a locally presentable symmetric monoidal closed category. Eg. Set ! John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems The basic context ◮ V a locally presentable symmetric monoidal closed category. Eg. Set ! ◮ E a locally presentable V -category and K : A ֒ → E a small dense full subcategory of arities . John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems The basic context ◮ V a locally presentable symmetric monoidal closed category. Eg. Set ! ◮ E a locally presentable V -category and K : A ֒ → E a small dense full subcategory of arities . ◮ Main examples I will talk about are when V = Set . John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems The basic context ◮ V a locally presentable symmetric monoidal closed category. Eg. Set ! ◮ E a locally presentable V -category and K : A ֒ → E a small dense full subcategory of arities . ◮ Main examples I will talk about are when V = Set . ◮ The K -nerve functor N K = E ( K − , 1) : E → [ A op , V ] is fully faithful. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems The basic context ◮ V a locally presentable symmetric monoidal closed category. Eg. Set ! ◮ E a locally presentable V -category and K : A ֒ → E a small dense full subcategory of arities . ◮ Main examples I will talk about are when V = Set . ◮ The K -nerve functor N K = E ( K − , 1) : E → [ A op , V ] is fully faithful. ◮ If X : A op → V is isomorphic to N K A we say that X is a K -nerve. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context ◮ E = Set and A = F the full subcategory of finite cardinals. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context ◮ E = Set and A = F the full subcategory of finite cardinals. ◮ K -nerves F op → Set ≡ finite product preserving functors. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context ◮ E = Set and A = F the full subcategory of finite cardinals. ◮ K -nerves F op → Set ≡ finite product preserving functors. ◮ E a locally finitely presentable category and K : E f → E the inclusion of the skeletal full subcategory of finitely presentable objects. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context ◮ E = Set and A = F the full subcategory of finite cardinals. ◮ K -nerves F op → Set ≡ finite product preserving functors. ◮ E a locally finitely presentable category and K : E f → E the inclusion of the skeletal full subcategory of finitely presentable objects. ◮ K -nerves E op → Set ≡ finite limit preserving functors. f John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context ◮ E = Set and A = F the full subcategory of finite cardinals. ◮ K -nerves F op → Set ≡ finite product preserving functors. ◮ E a locally finitely presentable category and K : E f → E the inclusion of the skeletal full subcategory of finitely presentable objects. ◮ K -nerves E op → Set ≡ finite limit preserving functors. f ◮ Standard kinds of examples – E the free cocompletion of A under some class of colimit. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context ◮ E = Set and A = F the full subcategory of finite cardinals. ◮ K -nerves F op → Set ≡ finite product preserving functors. ◮ E a locally finitely presentable category and K : E f → E the inclusion of the skeletal full subcategory of finitely presentable objects. ◮ K -nerves E op → Set ≡ finite limit preserving functors. f ◮ Standard kinds of examples – E the free cocompletion of A under some class of colimit. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context II ◮ E = Grph and A = ∆ 0 . Contains graphs � 1 � · · · � n [ n ] := 0 for n > 0. John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context II ◮ E = Grph and A = ∆ 0 . Contains graphs � 1 � · · · � n [ n ] := 0 for n > 0. ◮ K -nerves ∆ op 0 → Set ≡ functors sending the wide pushouts [ n ] ∼ = [1] + [0] [1] + [0] . . . + [0] [1] to wide pullbacks (Segal condition). John Bourke Monads and theories
Introduction Basic setting and two standard constructions Main theorems Examples of the basic context II ◮ E = Grph and A = ∆ 0 . Contains graphs � 1 � · · · � n [ n ] := 0 for n > 0. ◮ K -nerves ∆ op 0 → Set ≡ functors sending the wide pushouts [ n ] ∼ = [1] + [0] [1] + [0] . . . + [0] [1] to wide pullbacks (Segal condition). ◮ E = [ G op , Set ] the category of globular sets, and A = Θ 0 the full subcategory of globular cardinals. John Bourke Monads and theories
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