R. A. G. Seely

Locally cartesian closed categories and type theory

It is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/ A are cartesian closed. In such a category, the notion of a ‘generalized set’, for example an ‘ A -indexed set’, is represented by a morphism B → A of C, i.e. by an object of C/ A . The point about such a category C is that C is a C-indexed category, and more, is a hyper-doctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A , the predicates with a free variable of type A are morphisms into A , and ‘proofs’ are morphisms over A . We see here a certain ‘ambiguity’ between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A , which is a predicate over A ; a morphism 1 → A can be viewed either as an object of type A or as a proof of the proposition A .

Weakly distributive categories

There are many situations in logic, theoretical computer science, and category theory where two binary operations — one thought of as a (tensor) “product”, the other a “sum” — play a key role. In distributive and ∗-autonomous categories these operations can be regarded as, respectively, the and/or of traditional logic and the times/par of (multiplicative) linear logic. In the latter logic, however, the distributivity of product over sum is conspicuously absent: this paper studies a “linearization” of that distributivity which is present in both case. Furthermore, we show that this weak distributivity is precisely what is needed to model Gentzens cut rule (in the absence of other structural rules) and can be strengthened in two natural ways to generate full distributivity and ∗-autonomous categories.

Natural deduction and coherence for weakly distributive categories

Abstract This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the two-tensor structure ( times/par ) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansion-reduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion-reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories. The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending similar results for monoidal categories to include the treatment of the tensor units. Second, we extend these coherence results to the full theory of ∗-autonomous categories — providing a decision procedure for the maps of free symmetric ∗-autonomous categories. Third, we derive a conservative extension result for the passage from weakly distributive categories to ∗-autonomous categories. We show strong categorical conservativity, in the sense that the unit of the adjunction between weakly distributive and ∗-autonomous categories is fully faithful.

HYPERDOCTRINES, NATURAL DEDUCTION AND THE BECK CONDITION

In the late sixties F. W. LAWVERE showed that the logical connectives and quantifiers were examples of the categorical notion of adjointness. In [9] and [lo] he amplified this notion by a more thorough discussion of the structure of a hyperdoctrine, which had much of the flavour of intuitionistic logic with equality. In this eontext it was natural to “stratify” formulae and proofs according to the free variables occurring in them, a procedure later to become standard in categorical logic. (See MAKKAIREYES [12], FOURMAN [l], KOCK-REYES [7], for example.) In this paper, we make the relationship between hyperdoctrines and logic precise, showing that hyperdoctrines are naturally equivalent to first order intuitionistic theories with equality, where here “theory” is intended to include some proof theoretic structure, and not merely the notion of entailment. Moreover, we will show that this equivalence restricts to one giving a natural logical interpretation to the BECK (or CHEVALLEY) condition: in a given hyperdoctrine, the Beck condition for a pullback diagram is just the condition that the corresponding theory “recognizes ” the pull back.

Linearly distributive functors

Abstract This paper introduces a notion of “linear functor” between linearly distributive categories that is general enough to account for common structure in linear logic, such as the exponentials (!, ?), and the additives (product, coproduct), and yet when interpreted in the doctrine of ∗ -autonomous categories, gives the familiar notion of monoidal functor. We show that there is a bi-adjunction between the 2-categories of linearly distributive categories and linear functors, and of ∗ -autonomous categories and monoidal functors, given by the construction of the “nucleus” of a linearly distributive category. We develop a calculus of proof nets for linear functors, and show how linearity accounts for the essential coherence structure of the exponentials and the additives.

Differential categories

Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a ‘coalgebra modality’) and a differential combinator satisfying a number of coherence conditions. In such a category one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable). Although such categories do not necessarily arise from models of linear logic, one should think of this as replacing the usual dichotomy of linear vs. stable maps established for coherence spaces.After establishing the basic axioms, we give a number of examples. The most important example arises from a general construction, a comonad

Introduction to linear bicategories

S_\infty

Feedback for linearly distributive categories: traces and fixpoints

on the category of vector spaces. This comonad and associated differential operators fully capture the usual notion of derivatives of smooth maps. Finally, we derive additional properties of differential categories in certain special cases, especially when the comonad is a storage modality, as in linear logic. In particular, we introduce the notion of a categorical model of the differential calculus, and show that it captures the not-necessarily-closed fragment of Ehrhard–Regnier differential

A Hyperdoctrinal View of Concurrent Constraint Programming

\lambda

Categories for computation in context and unified logic

-calculus.