We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad—a structure with 2-cells of all possible double-categorical shapes—equipped with all possible composition operations, coherently. We also characterize them using “implicit” double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a “tidiness” condition to the double bicategories of Verity, or to the cubical bicategories of Garner.

We consider the problem of constructing the free bifibration generated by a functor of categories $𝑝:𝐷\to 𝐶$. This problem was previously considered by Lamarche, and is closely related to the problem, considered by Dawson, Paré, and Pronk, of “freely adjoining adjoints” to a category. We develop a proof-theoretic approach to the problem, beginning with a construction of the free bifibration ${\mathrm{\Lambda }}_{𝑝}:𝐵𝑖𝑓𝑖𝑏(𝑝)\to 𝐶$ in which objects of $𝐵𝑖𝑓𝑖𝑏(𝑝)$ are formulas of a primitive “bifibrational logic”, and arrows are derivations in a cut-free sequent calculus modulo a notion of permutation equivalence. We show that instantiating the construction to the identity functor generates a _zigzag double category_ $\mathbb{Z}(𝐶)$, which is also the free double category with companions and conjoints (or fibrant double category) on $𝐶$. The approach adapts smoothly to the more general task of building $(𝑃,𝑁)$-fibrations, where one only asks for pushforwards along arrows in $𝑃$ and pullbacks along arrows in $𝑁$ for some subsets of arrows; this encompasses Kock and Joyal’s notion of _ambifibration_ when $(𝑃,𝑁)$ form a factorization system. We establish a series of progressively stronger normal forms, guided by ideas of _focusing_ from proof theory, and obtain a canonicity result under assumption that the base category is factorization preordered relative to $𝑃$ and $𝑁$. This canonicity result allows us to decide the word problem and to enumerate relative homsets without duplicates. Finally, we describe several examples of a combinatorial nature, including a category of plane trees generated as a free bifibration over $𝜔$, and a category of increasing forests generated as a free ambifibration over $\mathrm{\Delta }$, which contains the lattices of noncrossing partitions as quotients of its fibers by the Beck-Chevalley condition for bicartesian squares.

We define strict and lax orthogonal factorization systems on double categories. These consist of an orthogonal factorization system on arrows and one on double cells that are compatible with each other. Our definitions are motivated by several explicit examples, including factorization systems on double categories of spans, relations and bimodules. We then prove monadicity results for orthogonal factorization systems on double categories in order to justify our definitions. For fibrant double categories we discuss the structure of the double orthogonal factorization systems that have a given orthogonal factorization system on the arrows in common. Finally, we study the interaction of orthogonal factorization systems on double categories with double fibrations.

We give a denotational account of logical relations for call-by-push-value (CBPV) in the fibrational style of Hermida, Jacobs, Katsumata and others. Fibrations – which axiomatise the usual notion of sets-with-relations – provide a clean framework for constructing new, logical relations-style, models. Such models can then be used to study properties such as effect simulation.

Extending this picture to CBPV is challenging: the models incorporate both adjunctions and enrichment, making the appropriate notion of fibration unclear. We handle this using 2-category theory. We identify an appropriate 2-category, and define CBPV fibrations to be fibrations internal to this 2-category which strictly preserve the CBPV semantics.

Next, we develop the theory so it parallels the classical setting. We give versions of the codomain and subobject fibrations, and show that new models can be constructed from old ones by pullback. The resulting framework enables the construction of new, logical relations-style, models for CBPV.

Finally, we demonstrate the utility of our approach with particular examples. These include a generalisation of Katsumata’s $\top \top$-lifting to CBPV models, an effect simulation result, and a relative full completeness result for CBPV without sum types.

Gradually typed programming languages, which allow for soundly mixing static and dynamically typed programming styles, present a strong challenge for metatheorists. Even the simplest sound gradually typed languages feature at least recursion and errors, with realistic languages featuring furthermore runtime allocation of memory locations and dynamic type tags. Further, the desired metatheoretic properties of gradually typed languages have become increasingly sophisticated: validity of type-based equational reasoning as well as the relational property known as graduality. Many recent works have tackled verifying these properties, but the resulting mathematical developments are highly repetitive and tedious, with few reusable theorems persisting across different developments.

In this work, we present a new denotational semantics for gradual typing developed using guarded domain theory. Guarded domain theory combines the generality of step-indexed logical relations for modeling advanced programming features with the modularity and reusability of denotational semantics. We demonstrate the feasibility of this approach with a model of a simple gradually typed lambda calculus and prove the validity of beta-eta equality and the graduality theorem for the denotational model. This model should provide the basis for a reusable mathematical theory of gradually typed program semantics. Finally, we have mechanized most of the core theorems of our development in Guarded Cubical Agda, a recent extension of Agda with support for the guarded recursive constructions we use.

Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product.

We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an ordered linear restriction on standard predicate logic, which guarantees that all functions between categories are functorial, all relations are profunctorial, and all transformations are natural by construction, with no separate proofs necessary. Important category-theoretic proofs such as the Yoneda lemma and Co-yoneda lemma become simple type-theoretic proofs about the relationship between unit, tensor and (ordered) function types, and can be seen to be ordered refinements of theorems in predicate logic. The type theory is sound and complete for a categorical model in virtual equipments, which model both internal and enriched category theory. While the proofs in our type theory look like standard set-based arguments, the syntactic discipline ensure that all proofs and constructions carry over to enriched and internal settings as well.

In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a network model, a monoidal variant of Joyal’s combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green’s graph products of groups to the context of universal algebra. For the latter, we examine the emergence of monoidal fibrations from the presence of catalysts in Petri nets.

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely lax monoidal pseudofunctors to the 2-category of categories. Furthermore, we investigate the relation between this ‘global’ monoidal version where the total category is monoidal and the fibration strictly preserves the structure, and a ‘fibrewise’ one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, we show how lax monoidal structures on a pseudofunctor to Cat bijectively correspond to lifts of the pseudofunctor to MonCat. Finally, we give some examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.

We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism, which were developed in blame calculi and to state the “gradual guarantee” theorem of gradual typing. Combined with the ordinary extensionality ($𝜂$) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of Findler and Felleisen’s definitions of contracts, and use it to build some concrete domain-theoretic models of gradual typing.

We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism. These dynamism judgements build on prior work in blame calculi and on the “gradual guarantee” theorem of gradual typing. Combined with the ordinary extensionality (eta) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs, which is a semantic analogue of the interpretation of gradual typing using contracts, and use it to build some concrete domain-theoretic models of gradual typing.

The concept of refinement in type theory is a way of reconciling the “intrinsic” and the “extrinsic” meanings of types. We begin with a rigorous analysis of this concept, settling on the simple conclusion that the type-theoretic notion of “type refinement system” may be identified with the category-theoretic notion of “functor”. We then use this correspondence to give an equivalent type-theoretic formulation of Grothendieck’s definition of (bi)fibration, and extend this to a definition of monoidal closed bifibrations, which we see as a natural space in which to study the properties of proofs and programs. Our main result is a representation theorem for strong monads on a monoidal closed fibration, describing sufficient conditions for a monad to be isomorphic to a continuations monad “up to pullback”.