Eilenberg-Kelly Reloaded Tarmo Uustalu, Reykjavik U. Niccol` o - - PowerPoint PPT Presentation
Eilenberg-Kelly Reloaded Tarmo Uustalu, Reykjavik U. Niccol` o Veltri, Tallinn U. of Techn. Noam Zeilberger, Ecole Polytechnique MFPS 2020, online talk Closed categories Closed categories [Eilenberg & Kelly 1966] are categories
◮ Closed categories [Eilenberg & Kelly 1966] are categories with a unit
◮ Examples:
◮ Categories of structured sets, e.g. normal bands, posets ◮ Categories underlying deductive systems, e.g. STLC
◮ In many cases, the internal hom is determined by an adjunction with
◮ Theorem: Given a category C equipped with a unit I and two
◮ Internal adjunction: the natural transformation
◮ Needed to invert associator αA,B,C : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C). ◮ Invertibility of α not matched by anything in defn. of closed category.
◮ [Street 2013] proposes a way to fix the mismatch: consider weak
◮ Left-skew monoidal categories [Szlach´
◮ Left-skew closed categories [Street 2013]
◮ Theorem: Given a category C equipped with a unit I and two
◮ No internal adjunction requirement!
◮ A left-skew monoidal category is a category C together with an
◮ N.B. λ, ρ, α are not required to be invertible.
◮ A left-skew closed category is a category C together with an object
◮ In the original definition of closed category, i and
◮ In a left-skew monoidal category, the invertibility of a structural law
◮ We identify analogous normality conditions in a left-skew closed
◮ Theorem: In the presence of an adjunction − ⊗ B ⊣ B ⊸ −, not
◮ The precise correspondence:
◮ In the original Eilenberg-Kelly theorem, the internal adjunction
◮ Changing the orientation of the structural laws ρ, λ, α of left-skew
A : A ⊗ I → A
A : A → I ⊗ A
A,B,C : A ⊗ (B ⊗ C) → (A ⊗ B) ⊗ C ◮ Similarly, changing the orientation of the structural laws i,
A,B : C(I, A ⊸ B) → C(A, B)
◮ We prove a right-skew variant of Street’s theorem connecting adjoint
◮ More interestingly, we prove a new theorem connecting left-skew
◮ Theorem: Let C be a category with an object I and functors
◮ The normality conditions on ⊸L and ⊸R also correspond:
◮ We discuss a large number of examples, in particular for motivating
◮ In this talk we discuss:
◮ Skewing a left-(right-)skew closed structure further to the left
◮ Lifting left- and right-skew closed structure to a Kleisli category. ◮ The non-commutative linear typed λ-calculus with unit type.
◮ Let (C, I, ⊸) be a left-skew closed category with a comonad D on it. ◮ Suppose D lax closed, i.e., coming with a map e : I → D I and a nat.
◮ Then C has another left-skew closed structure (I, D⊸) where
DiA = D I ⊸ A e⊸A
iA
◮ If both i and e are invertible, then Di is invertible. ◮ Instead, given (C, I, ⊸) right-skew closed and an oplax closed
◮ Let (C, I, ⊸) be a left-skew closed category with a monad T on it. ◮ Suppose T left-strong (or internally functorial), i.e., endowed with a
◮ Then Kl(T) has a left-skew closed structure (I, ⊸T) where
A = J ( I jA
A⊸ηA A ⊸ TA )
A = I ⊸ T A iTA
◮ If, instead, (C, I, ⊸) is right-skew closed and T is lax closed
◮ Types A, B ::= X | I | A ⊸ B, where X is an atomic type. ◮ Contexts are lists of types. ◮ Well-formed terms:
◮ Definitional equality of terms is βη-equality. ◮ It is a left-skew closed category. Derivation tree of L:
◮ This calculus is a concrete presentation of the free left-skew closed
◮ Fact:
◮ i becomes invertible if we replace the elimination rule for I with the
◮ Continuing work initiated by Street on a “cleaner” Eilenberg-Kelly
◮ We showed that closed categories (in the sense of the standard
◮ We also demonstrated that there is a well-justified notion of
◮ Future work:
◮ Find more examples relevant to mathematical semantics of
◮ In continuation to our prior work [UVZ 2018], develop the proof