Lawvere 2-theories joint work with John Power Stephen Lack - PowerPoint PPT Presentation

The classical case The general notion of Lawvere theory The two-dimensional case References Lawvere 2-theories joint work with John Power Stephen Lack University of Western Sydney 20 June 2007 Lawvere 2-theories Stephen Lack

The classical case The general notion of Lawvere theory The two-dimensional case References Outline The classical case The general notion of Lawvere theory The two-dimensional case References Lawvere 2-theories Stephen Lack

The classical case The general notion of Lawvere theory The two-dimensional case References Ordinary Lawvere theories ◮ Write S for the skeletal category of finite sete, and J : S → Set for the inclusion. S is the free category with finite coproducts/colimits on 1 ◮ A (classical) Lawvere theory is an identity-on-objects functor E : S op → L which preserves finite products/limits. L will have all finite products but not necessarily all finite limits. ◮ A model of L is a functor X : L → Set which preserves finite products ◮ Equivalently for which XE : S op → Set preserves finite products/limits . . . or equivalently for which XE = Set ( J − , A ) for some A ∈ Set (in fact A = X 1) ◮ L ( m , n ) = L ( m , 1) n , where L ( m , 1) is the set of m -ary operations Lawvere 2-theories Stephen Lack

� � � The classical case The general notion of Lawvere theory The two-dimensional case References The category of models ◮ Write Mod( L ) for the category of models; the morphisms are natural transformations ◮ Pullback diagram Mod( L ) � � [ L , Set ] [ E , Set ] U � [ S op , Set ] Set � � Set ( J , 1) where Set ( J , 1) sends a set X to corresponding finite-product-preserving functor Set ( J − , X ) : S op → Set ◮ Forgetful functor U is monadic; thus every Lawvere theory determines a monad on Set . Lawvere 2-theories Stephen Lack

� � � The classical case The general notion of Lawvere theory The two-dimensional case References Finitary monads on Set ◮ A functor is finitary if it preserves filtered colimits. A monad is finitary if its underlying endofunctor is so. T : Set → Set is finitary iff it is the left Kan extension of TJ : S → Set . Monads arising from Lawvere theories are finitary. ◮ Given a finitary monad T can form H � Set T L op � � � F T E S � � Set J and now E : S op → L is a Lawvere theory, and Set T is its category of models ◮ This gives an equivalence between Lawvere theories and finitary monads on Set [Linton]. Lawvere 2-theories Stephen Lack

The classical case The general notion of Lawvere theory The two-dimensional case References The enriched version (Power) ◮ Version involving symmetric monoidal closed V in place of Set ◮ V should be locally finitely presentable as a closed category (Kelly) in order to have good notion of finite object of V (i.e. arity). Then use finite cotensors in place of finite products ◮ For V -category K , object A ∈ K and X ∈ V , the cotensor A X (sometimes called X ⋔ A ) defined by K ( B , A X ) ∼ = V ( X , K ( B , A )) Say that K has finite cotensors if A X exists for all A ∈ K and all finitely presentable X ∈ V ◮ e.g. if V = Cat , can have operations with arity given by any finitely presentable category not just the discrete ones Lawvere 2-theories Stephen Lack

� � � The classical case The general notion of Lawvere theory The two-dimensional case References General notion of theory ◮ Consider a symmetric monoidal closed LFP V as above and an LFP V -category K i.e. K ≃ Lex( K op f , V ) for J : K f → K the full subcategory of finitely presentable objects. Want notion of theory equivalent to finitary V -monads on K ◮ Given finitary V -monad T , follow previous construction H � K T L op � � � F T E K f � � K J ◮ J preserves finite colimits, F T preserves colimits, and H reflects colimits, so E preserves finite colimits Definition (Nishizawa-Power) A Lawvere K -theory is an identity-on-objects, finite-limit-preserving E : K op � � L f Lawvere 2-theories Stephen Lack

� � � � � � The classical case The general notion of Lawvere theory The two-dimensional case References General notion of model ◮ Monad T and induced theory E : K op � � L as above f ◮ Pullback diagram K T ( H , 1) K T � � [ L , V ] U T [ E , V ] K ( J , 1) � [ K op K � � f , V ] Definition (Nishizawa-Power) The category of models of a theory E : K op � � L is given by f the pullback Mod( L ) � � [ L , V ] [ E , V ] K ( J , 1) � [ K op K � � f , V ] Lawvere 2-theories Stephen Lack

� � � � � � � The classical case The general notion of Lawvere theory The two-dimensional case References The equivalence between monads and theories Theorem (Nishizawa-Power) The category Law ( K ) of Lawvere theories on K is equivalent to the category Mnd f ( K ) of finitary monads on K . ◮ finitary monad T gives theory Φ( T ) given by E : K op � � L f ◮ theory E : K op � � L gives finitarily monadic Mod( L ) → K f and so finitary monad Ψ( L ) Ψ(Φ( T )) ∼ Φ(Ψ( L )) ∼ = T follows = L because Lan E from pullback gives free models Y L op � � � Mod( L ) � � � [ L , V ] K T � � [ L , V ] � � � [ E , V ] Lan E E F � [ K op � K � [ K op f , V ] K � � K f � � � � f , V ] Lawvere 2-theories Stephen Lack

� � � � � � The classical case The general notion of Lawvere theory The two-dimensional case References Models in other categories ◮ Since K ≃ Lex( K op f , V ) we can equivalently define models via the pullback Mod( L ) � � [ L , V ] [ E , V ] Lex( K op � [ K op f , V ) � � f , V ] ◮ If A has finite limits, then define category of models in A by the pullback Mod( L , A ) � � [ L , A ] [ E , A ] Lex( K op � [ K op f , A ) � � f , A ] ◮ Thus a model is a functor X : L → A whose restriction XE : K op → A along E preserves finite limits f Lawvere 2-theories Stephen Lack

The classical case The general notion of Lawvere theory The two-dimensional case References Models as right adjoint functors ◮ E : K op � � L theory f ◮ A category with limits ◮ We’ll consider models of L in A � A L model X � A L X is a model � A op � Mod( L ) L op A op iff right adj. � A op [ L , V ] A (1 , X ) : A op → [ L , V ] � A op Mod( L ) left adj. left adj. lands in Mod( L ). A op A (1 , X ) � [ L , V ] Mod( L ) op � A right adj. right adj. ◮ and so Mod( L , A ) ≃ Radj(Mod( L ) op , A ) (cf Kelly notion of comodel wrt dense functor.) Lawvere 2-theories Stephen Lack

The classical case The general notion of Lawvere theory The two-dimensional case References The theory of theories ◮ K is LFP V -category ◮ |K f | set of objects of (a skeleton of) K f ◮ Forgetful functor Law ( K ) → [ |K f | 2 , V ] sending L to � � L ( Ec , Ed ) c , d ∈|K f | is finitarily monadic, and so Law ( K ) is LFP and is the category of models of a Lawvere theory in [ |K f | 2 , V ]. (cf Lack theorem on monadicity of Mnd f ( K ) over [ |K f | , K ].) Lawvere 2-theories Stephen Lack

The classical case The general notion of Lawvere theory The two-dimensional case References The Cat -enriched case ◮ Take V = Cat . ◮ Finitary 2-monads on an LFP 2-category K are equivalent to Lawvere 2-theories E : K op → L f ◮ Can describe such structures as monoidal category, category with limits and/or colimits of some type, categories with limits and colimits and exactness conditions, category with two monoidal structures and a distributive law, category with a factorization system, pair of monoidal categories with a monoidal adjunction etc. ◮ Need to weaken this to get lax and pseudo versions for example, want functors which preserve limits in the usual sense, not on-the-nose; and want monoidal or strong monoidal functors, not strict ones Lawvere 2-theories Stephen Lack

� � � � � � The classical case The general notion of Lawvere theory The two-dimensional case References Pseudomorphisms ◮ For 2-category C , write Ps( C , Cat ) for the 2-category of (strict) 2-functors, pseudonatural transformations, and modifications and [ C , Cat ] for sub-2-category of strict maps ◮ If K LFP have maps K → [ K op f , Cat ] → Ps( K op f , Cat ) ◮ For theory L define 2-category Mod( L ) ps of strict models and pseudomaps by pullback Mod( L ) ps Ps( L , Cat ) Ps( E , Cat ) � Ps( K op f , Cat ) K ◮ So a pseudomap between models X , X ′ : L → Cat is a pseudonatural transformation f : X � X ′ with fE strict: X K op E � � L � � Cat � � f f X ′ Lawvere 2-theories Stephen Lack

� � � The classical case The general notion of Lawvere theory The two-dimensional case References More on pseudomorphisms ◮ for a model X , we have XC = ( X 1) C where X 1 is the underlying object ◮ L ( C , 1) is the category of C -ary operations ◮ for γ : C → 1 in L , have operation X γ : ( X 1) C → X 1 ◮ for a pseudomorphism f : X → Y have pseudonaturality isomorphisms X γ ( X 1) C X 1 ∼ ( f 1) C = f 1 � Y 1 ( Y 1) C Y γ which show how f preserves the operation γ up to isomorphism Lawvere 2-theories Stephen Lack