Profinite Monads and Reitermans Theorem J. Ad amek, L.-T. Chen, S. - PowerPoint PPT Presentation

Profinite Monads and Reiterman’s Theorem J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Category Theory 2019 Edinburg J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 1

� � The Birkhoff Variety Theorem (1935) The Birkhoff Theorem A a full subcategory of Σ- Alg : A presentable by equations ⇔ variety (= HSP class) � ❍❍❍❍❍❍❍❍❍ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ regular quotients subobjects products J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 2

� � The Birkhoff Variety Theorem (1935) The Birkhoff Theorem A a full subcategory of Σ- Alg : A presentable by equations ⇔ variety (= HSP class) � ❍❍❍❍❍❍❍❍❍ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ regular quotients subobjects products Lawvere: equations are pairs of n . t . α : U n → U for U : Σ- Alg → Set An algebra A satisfies α = α ′ iff α A = α ′ A J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 2

The Reiterman Theorem (1982) The Reiterman Theorem � � A a full subcategory of Σ- Alg f : A presentable by pseudoequations ⇔ pseudovariety (= HSP f class) J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 3

The Reiterman Theorem (1982) The Reiterman Theorem � � A a full subcategory of Σ- Alg f : A presentable by pseudoequations ⇔ pseudovariety (= HSP f class) � � U f : Σ- Alg f → Set f Pseudoequations are pairs of n . t . α : U n f → U f a finite algebra A satisfies α = α ′ iff α A = α ′ A J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 3

The Reiterman Theorem (1982) Example Un , unary algebras σ : A → A A finite ⇒ ∃ n : σ n = ( σ n ) 2 Notation : σ ∗ = σ n Pseudoequation : σ ∗ ( x ) = x presents : finite algebras with σ invertible J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 4

� � Banaschewski and Herrlich (1976) D a complete category ( E , M ) a proper factorization system (e.g. regular epi - mono) notation ։ and ֌ D has enough projectives X : ∀ D ∃ X ։ D Definitions An equation e : X ։ A , X projective. ∀ f It is satisfied by D ∈ D if X � � D ❄ ❄ ⑧ ❄ ❄ ⑧ ❄ ❄ e ⑧ ❄ ∃ ❄ ⑧ A ( D is e -injective) Theorem A full subcategory A of D : A presentable by equations ⇔ a variety (= HSP class) J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 5

Pseudovariety Presentation Assume : D and ( E , M ) as above D f ⊆ D full subcategory closed under S and P f ’finite’ objects J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 6

Pseudovariety Presentation Assume : D and ( E , M ) as above D f ⊆ D full subcategory closed under S and P f ’finite’ objects Definition A pseudovariety is a full subcategory of D f closed under HSP f . J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 6

� �� � � � � Pseudovariety Presentation Definition A quasi-equation over X (projective) is a semilattice Ω of finite quotients e : X ։ A ( A ∈ D f ) X ✾ � ✝✝✝✝✝✝✝ ✾ ✾ e ′ e ✾ ✾ e ¯ ✾ ✾ � � ∀ e , e ′ ∈ Ω ¯ A ′ A ′′ u ′ � A ✽ u ′′ ✽ � ✆✆✆✆✆✆✆ ✽ ✽ ✽ ✽ ✽ e = e ∧ e ′ ∈ Ω A ′ × A ′′ ¯ J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 7

� �� � � � � � Pseudovariety Presentation Definition A quasi-equation over X (projective) is a semilattice Ω of finite quotients e : X ։ A ( A ∈ D f ) X ✾ � ✝✝✝✝✝✝✝ ✾ ✾ e ′ e ✾ ✾ e ¯ ✾ ✾ � � ∀ e , e ′ ∈ Ω ¯ A ′ A ′′ u ′ � A ✽ u ′′ ✽ � ✆✆✆✆✆✆✆ ✽ ✽ ✽ ✽ ✽ e = e ∧ e ′ ∈ Ω A ′ × A ′′ ¯ ∀ f � D An object D satisfies Ω if it is injective: X ❄ ⑧ ❄ ⑧ ❄ ⑧ ∃ e ∈ Ω ∃ ❄ � � ⑧ A J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 7

Pseudovariety Presentation Proposition A a full subcategory of D f : A presentable by quasi-equations ⇔ A a pseudovariety J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 8

� �� Pseudovariety Presentation Proposition A a full subcategory of D f : A presentable by quasi-equations ⇔ A a pseudovariety Proof ⇐ For every X projective Ω X : X ։ A ( A ∈ A ) Ω X semilattice ⇐ A is SP f -class D ∈ A ⇒ D satisfies Ω X . . . trivial f � � D D satisfies each Ω X ⇒ D ∈ A : choose X , ⑦ ⑦ ⑦ e ⑦ A X projective, e ∈ E , A ∈ A ⇒ D ∈ A J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 8

Our Goal Given : D , ( E , M ) and D f as above T a monad on D preserving E Describe pseudovarieties in D T by equations in some extension of D T J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 9

Our Goal Given : D , ( E , M ) and D f as above T a monad on D preserving E Describe pseudovarieties in D T by equations in some extension of D T D T has the factorization system inherited from D it has enough projectives : ( TX , µ X ) with X projective def D T = all algebras ( A , α ) with A ∈ D f f J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 9

Our Goal Given : D , ( E , M ) and D f as above T a monad on D preserving E Describe pseudovarieties in D T by equations in some extension of D T D T has the factorization system inherited from D it has enough projectives : ( TX , µ X ) with X projective def D T = all algebras ( A , α ) with A ∈ D f f Thus pseudovarieties are presentable by quasi-equations in D T J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 9

The Category ˆ D f Profinite completion Pro D f = ˆ D f (dual to Ind) finitely complete ⇒ ˆ D f complete D f ˆ E = cofiltered limits of quotients in D f ˆ M = cofiltered limits of subobjects in D f J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 10

The Category ˆ D f Profinite completion Pro D f = ˆ D f (dual to Ind) finitely complete ⇒ ˆ D f complete D f ˆ E = cofiltered limits of quotients in D f ˆ M = cofiltered limits of subobjects in D f Wanted : ˆ D f has enough ˆ E -projectives T yields (canonically) a monad ˆ T on ˆ D f preserving ˆ E ⇒ ˆ D f , ( ˆ E , ˆ M ) and ˆ T satisfy all of our assumptions Goal : quasi-equations in D T ⇔ equations in ( ˆ D f ) ˆ T Important : T and ˆ T have the same finite algebras ˆ f ≃ ˆ D T T D f J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 10

Profinite Factorization Systems Definition ( E , M ) is a profinite factorization system if E is closed under cofiltered limits of quotients in D → f Examples with E = surjective morphisms ˆ Set : Set f = Stone J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 11

Profinite Factorization Systems Definition ( E , M ) is a profinite factorization system if E is closed under cofiltered limits of quotients in D → f Examples with E = surjective morphisms ˆ Set : Set f = Stone Pos : with E = surjective monotone maps ˆ Pos f = Priestley J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 11

Profinite Factorization Systems Definition ( E , M ) is a profinite factorization system if E is closed under cofiltered limits of quotients in D → f Examples with E = surjective morphisms ˆ Set : Set f = Stone Pos : with E = surjective monotone maps ˆ Pos f = Priestley D ⊆ Σ- Str full subcategory closed under limits arbitrary operation symbols + finitely many relation symbols Pro D f ⊆ Stone D ˆ E = surjective continuous homomorphisms J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 11

Profinite monad ˆ T ˆ f → ˆ T is the codensity monad of the forgetful functor D T D f J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 12

Profinite monad ˆ T ˆ f → ˆ T is the codensity monad of the forgetful functor D T D f Example D = Set , TX = X ∗ : the word monad ˆ ˆ T is the monad of profinite words on Mon f = Stone Mon ˆ TY is the cofiltered limit of all finite ˆ E -quotients of Y carried by T -algebras Example For TX = X ∗ : a profinite word in a Stone monoid Y is a compatible choice of a member of A for every finite quotient monoid A of Y . J. Ad´ amek, L.-T. Chen, S. Milius and H. Urbat Profinite Monads and Reiterman’s Theorem 12