arXiv subjects
Publications and source records attributed to Richard Garner.
We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak omega-categories, showing that the universal and canonical cofibrant replacement of the operad for strict omega-categories is precisely Leinster's operad for weak omega-categories.
We describe a non-extensional variant of Martin-Löf type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.
We define a notion of weak omega-category internal to a model of Martin-Löf type theory, and prove that each type bears a canonical weak omega-category structure obtained from the tower of iterated identity types over that type. We show that the omega-categories arising in this way are in fact omega-groupoids.
An algebraically exact category in one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small limits and sifted colimits for which the functor taking sifted colimits is continuous. They conjectured that a complete and sifted-cocomplete category should be algebraically exact just when it is Barr-exact, finite limits commute with filtered colimits, regular epimorphisms are stable by small products, and filtered colimits distribute over small products. We prove this conjecture.
We define a notion of symmetric monoidal closed (SMC) theory, consisting of a SMC signature augmented with equations, and describe the classifying categories of such theories in terms of proof nets.
This paper investigates the use of symmetric monoidal closed (SMC) structure for representing syntax with variable binding, in particular for languages with linear aspects. In our setting, one first specifies an SMC theory T, which may express binding operations, in a way reminiscent from higher-order abstract syntax. This theory generates an SMC category S(T) whose morphisms are, in a sense, terms in the desired syntax. We apply our approach to Jensen and Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This leads to an alternative category of bigraphs, which we compare to the original.
We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids.
In this paper, we give a novel abstract description of Szabo's polycategories. We use the theory of double clubs -- a generalisation of Kelly's theory of clubs to `pseudo' (or `weak') double categories -- to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the `two-sided Kleisli bicategory' of this pseudo-distributive law are precisely symmetric polycategories.
The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.
There is an ``algebraisation'' of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of maps-with-structure, where the extra structure on a map now encodes a choice of liftings with respect to the other class. This extra structure has pleasant consequences: for example, a natural w.f.s. on C induces a canonical natural w.f.s. structure on any functor category [A, C]. In this paper, we define cofibrantly generated natural weak factorisation systems by analogy with cofibrantly generated w.f.s.'s. We then construct them by a method which is reminiscent of Quillen's small object argument but produces factorisations which are much smaller and easier to handle, and show that the resultant natural w.f.s. is, in a suitable sense, freely generated by its generating cofibrations. Finally, we show that the two categories of maps-with-structure for a natural w.f.s. are closed under all the constructions we would expect of them: (co)limits, pushouts / pullbacks, transfinite composition, and so on.
We develop a theory of double clubs which extends Kelly's theory of clubs to the pseudo double categories of Pare and Grandis. We then show that the club for symmetric strict monoidal categories on Cat extends to a `double club' on the pseudo double category of `categories, functors, profunctors and transformations'.