Eilenberg-Kelly Reloaded

Abstract

The Eilenberg-Kelly theorem states that a category $C$ with an object I and two functors $\otimes :C\times C\to C$ and $⊸:C^{\mathrm{op}}\times C\to C$ related by an adjunction $-\otimes B⊣B⊸-$ natural in B is monoidal iff it is closed and moreover the adjunction holds internally. We dissect the proof of this theorem and observe that the necessity for a side condition on closedness arises because the standard definition of closed category is left-skew in regards to associativity. We analyze Street's observation that left-skew monoidality is equivalent to left-skew closedness and establish that monoidality is equivalent to closedness unconditionally under an adjusted definition of closedness that requires normal associativity. We also work out a definition of right-skew closedness equivalent to right-skew monoidality. We give examples of each type of structure; in particular, we look at the Kleisli category of a left-strong monad on a left-skew closed category and the Kleisli category of a lax closed monad on a right-skew closed category. We also view skew and normal monoidal and closed categories as special cases of skew and normal promonoidal categories and take a brief look at left-skew prounital-closed categories.