Linear Logic without Units

We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of pro-monoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this – and the simplest, once the (substantial) groundwork has been laid – is to consider promonoidal categories as an instance of the general theory of pseudomonoids in a monoidal bicategory. Accordingly, we describe and explain the notions of monoidal bicategory and pseudomonoid therein. The higher-dimensional nature of monoidal bicategories presents serious no-tational difficulties, since to use the natural analogue of the commutative diagrams used in ordinary category theory would require the use of three-dimensional diagrams. We therefore introduce a novel technical device, which we dub the calculus of components, that dramatically simplifies the business of reasoning about a certain class of algebraic structure internal to a monoidal bicate-gory. When viewed through this simplifying lens, the theory of pseudomonoids turns out to be essentially formally identical to the ordinary theory of monoidal categories – at least in the absence of permutative structure such as braiding or symmetry. We indicate how the calculus of components may be extended to cover structures that make use of the braiding in a braided monoidal bicate-gory, and use this to study braided pseudomonoids. A higher-dimensional analogue of Cayley's theorem is proved, and used to deduce a novel characterisation of the unit of a promonoidal category. This, and the other preceding work, is then used to give two characterisations of the categories that model the unitless fragment of intuitionistic multiplicative linear logic. Finally we consider the non-intuitionistic case, where the second charac-terisation in particular takes a surprisingly simple form.  Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning.  Copyright The author of this thesis (including any appendices and/or schedules to this thesis) owns any copyright in it (the " Copyright ") and he has given The University of Manchester the right to use such Copyright for any administrative, promotional , educational and/or teaching purposes. Copies of this thesis, either in full or in extracts, may be made only in accordance with …