SLIDE 1 Distributive Laws1
Liang Ze Wong
University of Washington, Seattle
Category Theory 2017
1Strings attached!
SLIDE 2 Outline
- 1. String diagrams, monads, adjunctions
- 2. Distributive laws between monads S, T
- 3. Lifts of monads T to the category of algebras XS
- 4. (2) ⇐
⇒ (3)
SLIDE 3
String diagrams for monads
µ η X T T T T T : X → X µ: TT ⇒ T η: 1X ⇒ T
SLIDE 4
String diagrams for adjunctions
F : X → Y U : Y → X η: 1X ⇒ UF ε: FU ⇒ 1Y ε η X Y F U F ε η U F U Y
SLIDE 5
Adjunctions give monads give adjunctions
T := UF µT := UεF ηT := η T := U F :=
SLIDE 6
Adjunctions give monads give adjunctions
T := UF µT := UεF ηT := η T := U F := Y := XT F := F T U := UT
SLIDE 7
Distributive laws
Definition
Let S, T be monads on X. A distributive law of S over T is a natural transformation ℓ: ST ⇒ TS S T ℓ such that ...
SLIDE 8
Distributive laws
= =
SLIDE 9
Distributive laws
= =
SLIDE 10
Distributive laws
A distributive law of S over T makes TS a monad: But today we’ll look at a different characterization.
SLIDE 11
Lifts of monads
Definition
Let S, T be monads on X. A lift of T to XS is a monad ( ˜ T, ˜ µT, ˜ ηT) on XS such that US ˜ T = TUS US ˜ µT = µTUS US ˜ ηT = ηTUS
SLIDE 12
Lifts of monads
Definition
Let S, T be monads on X. A lift of T to XS is a monad ( ˜ T, ˜ µT, ˜ ηT) on XS such that US ˜ T = TUS US ˜ µT = µTUS US ˜ ηT = ηTUS XS XS X X
˜ T US US T
SLIDE 13
Lifts of monads
Definition
Let S, T be monads on X. A lift of T to XS is a monad ( ˜ T, ˜ µT, ˜ ηT) on XS such that US ˜ T = TUS US ˜ µT = µTUS US ˜ ηT = ηTUS XS XS X X
˜ T US US T
T US ˜ T US ˜ T T
SLIDE 14
Lifts give distributive laws
Lemma
Let S, T be monads on X such that T lifts to a monad ˜ T on XS. Then there is a distributive law of S over T.
SLIDE 15
Lifts give distributive laws
Lemma
Let S, T be monads on X such that T lifts to a monad ˜ T on XS. Then there is a distributive law of S over T.
SLIDE 16
Lifts give distributive laws
Lemma
Let S, T be monads on X such that T lifts to a monad ˜ T on XS. Then there is a distributive law of S over T. :=
SLIDE 17
Lifts give distributive laws
Lemma
Let S, T be monads on X such that T lifts to a monad ˜ T on XS. Then there is a distributive law of S over T. :=
SLIDE 18
Lifts give distributive laws
Lemma
Let S, T be monads on X such that T lifts to a monad ˜ T on XS. Then there is a distributive law of S over T. := Note: This can be done with lifts over any adjunction yielding S.
SLIDE 19
Distributive laws give lifts
Lemma
Suppose there is a distributive law ℓ: ST ⇒ TS of S over T. Then T lifts to a monad ˜ T over XS.
SLIDE 20 Distributive laws give lifts
Lemma
Suppose there is a distributive law ℓ: ST ⇒ TS of S over T. Then T lifts to a monad ˜ T over XS. This requires the universal property of XS:
G : Y → XS ∼ =
with S-action σ : SG ⇒ G
XS X
˜ G G US
SLIDE 21 Distributive laws give lifts
Lemma
Suppose there is a distributive law ℓ: ST ⇒ TS of S over T. Then T lifts to a monad ˜ T over XS. This requires the universal property of XS:
G : Y → XS ∼ =
with S-action σ : SG ⇒ G
XS X
˜ T ? US
SLIDE 22
Distributive laws give lifts
XS XS X X
˜ T US US T
SLIDE 23
Distributive laws give lifts
XS XS X X
˜ T US US T
XS US T
SLIDE 24
Distributive laws give lifts
XS XS X X
˜ T US US T
XS US T S
SLIDE 25
=
SLIDE 26
=
Thank you!
Questions?
SLIDE 27
References
◮ Jon Beck. Distributive laws.
Seminar on triples and categorical homology theory, 119–140. Springer, 1969.
◮ Eugenia Cheng. Distributive laws for Lawvere theories.
arXiv:1112.3076, 2011.
◮ Eugenia Cheng. Distributive laws 1-4 (videos).
https://www.youtube.com/playlist?list= PLEC25F0F5AC915192
◮ Ross Street. The formal theory of monads.
Journal of Pure and Applied Algebra, 2(2):149–168, 1972.
SLIDE 28
Distributive law to lift to distributive law
◮ Start with a distributive law ◮ This gives a lift satisfying
=
◮ Using the lift, define another distributive law. Check that this
is the same as the one we started with: = =
SLIDE 29
Lift to distributive law to lift
◮ Starting with a lift, define a distributive law ◮ This gives another lift of T, which also precomposes with US
to yield TUS.
◮ To check that they are the same lift, need to check that the
induced S-actions on TUS are the same: = =
