  Monoidal closed categories naturally model NMILL, non-commutative
multiplicative intuitionistic linear logic: the monoidal unit and tensor
interpret the multiplicative verum and conjunction; the internal hom interprets
linear implication. In recent years, the weaker notion of (left) skew monoidal
closed category has been proposed by Ross Street, where the three structural
laws of left and right unitality and associativity are not required to be
invertible, they are merely natural transformations with a specific
orientation. A question arises: is it possible to find a logic which is
naturally modelled by skew monoidal closed categories? We answer positively by
introducing a cut-free sequent calculus for a skew version of NMILL that is a
presentation of the free skew monoidal closed category. We study the
proof-theoretic semantics of the sequent calculus by identifying a calculus of
derivations in normal form, obtained from an adaptation of Andreoli's focusing
technique to the skew setting. The resulting focused sequent calculus
peculiarly employs a system of tags for keeping track of new formulae appearing
in the antecedent and appropriately reducing non-deterministic choices in proof
search. Focusing solves the coherence problem for skew monoidal closed
categories by exhibiting an effective procedure for deciding equality of maps
in the free such category.
