We develop fibrational perspectives on context-free grammars and on nondeterministic finite-state automata over categories and operads. A generalized CFG is a functor from a free colored operad (aka multicategory) generated by a pointed finite species into an arbitrary base operad: this encompasses classical CFGs by taking the base to be a certain operad constructed from a free monoid, as an instance of a more general construction of an \emph{operad of spliced arrows} $\mathcal{W}\,\mathcal{C}$ for any category $\mathcal{C}$. A generalized NFA is a functor from an arbitrary bipointed category or pointed operad satisfying the unique lifting of factorizations and finite fiber properties: this encompasses classical word automata and tree automata without $\epsilon$-transitions, but also automata over non-free categories and operads. We show that generalized context-free and regular languages satisfy suitable generalizations of many of the usual closure properties, and in particular we give a simple conceptual proof that context-free languages are closed under intersection with regular languages. Finally, we observe that the splicing functor $\mathcal{W} : Cat \to Oper$ admits a left adjoint $\mathcal{C}: Oper \to Cat$, which we call the \emph{contour category} construction since the arrows of $\mathcal{C}\,\mathcal{O}$ have a geometric interpretation as oriented contours of operations of $\mathcal{O}$. A direct consequence of the contour / splicing adjunction is that every pointed finite species induces a universal CFG generating a language of \emph{tree contour words.} This leads us to a generalization of the Chomsky-Sch\"utzenberger Representation Theorem, establishing that a subset of a homset $L \subseteq \mathcal{C}(A,B)$ is a CFL of arrows if and only if it is a functorial image of the intersection of a $\mathcal{C}$-chromatic tree contour language with a regular language.

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.

The main goal of this article is to expose and relate different ways of interpreting the multiplicative fragment of classical linear logic in polycategories. Polycategories are known to give rise to models of classical linear logic in so-called representable polycategories with duals, which ask for the existence of various polymaps satisfying the different universal properties needed to define tensor, par, and negation. We begin by explaining how these different universal properties can all be seen as instances of a single notion of universality of a polymap parameterised by an input or output object, which also generalises the classical notion of universal multimap in a multicategory. We then proceed to introduce a definition of in-cartesian and out-cartesian polymaps relative to a refinement system (= strict functor) of polycategories, in such a way that universal polymaps can be understood as a special case. In particular, we obtain that a polycategory is a representable polycategory with duals if and only if it is bifibred over the terminal polycategory 1. Finally, we present a Grothendieck correspondence between bifibrations of polycategories and pseudofunctors into MAdj, the (weak) 2-polycategory of multivariable adjunctions. When restricted to bifibrations over 1 we get back the correspondence between *-autonomous categories and Frobenius pseudomonoids in MAdj that was recently observed by Shulman.

The Eilenberg-Kelly theorem states that a category C with an object I and two functors ⊗:C×C→C and ⊸:Cop×C→C related by an adjunction −⊗B⊣B⊸− natural in B is monoidal iff it is closed and moreover the adjunction holds internally. We dissect the proof of this theorem and observe that the necessity for a side condition on closedness arises because the standard definition of closed category is left-skew in regards to associativity. We analyze Street's observation that left-skew monoidality is equivalent to left-skew closedness and establish that monoidality is equivalent to closedness unconditionally under an adjusted definition of closedness that requires normal associativity. We also work out a definition of right-skew closedness equivalent to right-skew monoidality. We give examples of each type of structure; in particular, we look at the Kleisli category of a left-strong monad on a left-skew closed category and the Kleisli category of a lax closed monad on a right-skew closed category. We also view skew and normal monoidal and closed categories as special cases of skew and normal promonoidal categories and take a brief look at left-skew prounital-closed categories.

We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, right rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. We then describe two main applications of the coherence theorem, including: 1. A new proof of the lattice property for the Tamari order, and 2. A new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice $Y_n$.

Building on recently established enumerative connections between lambda calculus and the theory of embedded graphs (or "maps"), this paper develops an analogy between typing (of lambda terms) and coloring (of maps). Our starting point is the classical notion of an abelian group-valued "flow" on an abstract graph (Tutte, 1954). Typing a linear lambda term may be naturally seen as constructing a flow (on an embedded 3-valent graph with boundary) valued in a more general algebraic structure consisting of a preordered set equipped with an "implication" operation and unit satisfying composition, identity, and unit laws. Interesting questions and results from the theory of flows (such as the existence of nowhere-zero flows) may then be re-examined from the standpoint of lambda calculus and logic. For example, we give a characterization of when the local flow relations (across vertices) may be categorically lifted to a global flow relation (across the boundary), proving that this holds just in case the underlying map has the orientation of a lambda term. We also develop a basic theory of rewriting of flows that suggests topological meanings for classical completeness results in combinatory logic, and introduce a polarized notion of flow, which draws connections to the theory of proof-nets in linear logic and to bidirectional typing.

Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming andcontext effectsin linear logic, we study logical aspects of this ‘positive’ (covariant) representation, as well as of an associated ‘negative’ (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day's embedding theorem for monoidal closed categories. Then, we establish that the positive and negative representations satisfy an Isbell-style duality. As corollaries, we derive two different formulas for the positive representation of a pushforward (inspired by the classical negative translations of proof theory), which express it either as the dual of a pullback of a dual or as the double dual of a pushforward. Besides explaining how these constructions on refinement systems generalize familiar category-theoretic ones (by viewing categories as special refinement systems), our main running examples involve representations of Hoare logic and linear sequent calculus.

Combining insights from the study of type refinement systems and of monoidal closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of presheaves using a full and faithful embedding into a monoidal closed bifibration living now over the compact closed category of small categories and distributors. Besides revealing dualities which are not immediately apparent in the traditional presentation of the presheaf hyperdoctrine, this reconstruction leads us to an axiomatic treatment of directed equality predicates (modelled by hom presheaves), realizing a vision initially set out by Lawvere (1970). It also leads to a simple calculus of string diagrams (representing presheaves) that is highly reminiscent of C. S. Peirce's existential graphs for predicate logic, refining an earlier interpretation of existential graphs in terms of Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this work extends to a bifibrational setting a number of fundamental ideas of linear logic.

Abstract The main aim of the paper is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between α-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact-oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.

The standard reading of type theory through the lens of category theory is based on the idea of viewing a type system as a category of well-typed terms. We propose a basic revision of this reading: rather than interpreting type systems as categories, we describe them as functors from a category of typing derivations to a category of underlying terms. Then, turning this around, we explain how in fact any functor gives rise to a generalized type system, with an abstract notion of typing judgment, typing derivations and typing rules. This leads to a purely categorical reformulation of various natural classes of type systems as natural classes of functors.

Polarized logic is the logic of values and continuations, and their interaction through continuation-passing style. The main limitations of this logic are the limitations of CPS: that continuations cannot be composed, and that programs are fully sequentialized. Delimited control operators were invented in response to the limitations of classical continuation-passing. That suggests the question: what is the logic of delimited continuations? We offer a simple account of delimited control, through a natural generalization of the classical notion of polarity. This amounts to breaking the perfect symmetry between positive and negative polarity in the following way: answer types are positive. Despite this asymmetry, we retain all of the classical polarized connectives, and can explain "intuitionistic polarity'' (e.g., in systems like CBPV) as a restriction on the use of connectives, i.e., as a logical fragment. Our analysis complements and generalizes existing accounts of delimited control operators, while giving us a rich logical language through which to understand the interaction of control with monadic effects.

Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the Curry-Howard interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with a notion of scoped inference rules, which are used to represent binding. On the other hand, the usual computational arrow classifies recursive functions defined by pattern-matching. Unlike many previous approaches, both kinds of implication are connectives in a single logic, which serves as a rich logical framework capable of representing inference rules that mix binding and computation.

Focusing is a proof-search strategy, originating in linear logic, that elegantly eliminates inessential nondeterminism, with one byproduct being a correspondence between focusing proofs and programs with explicit evaluation order. Higher-order abstract syntax (HOAS) is a technique for representing higher-order programming language constructs (e.g., λ's) by higher-order terms at the"meta-level", thereby avoiding some of the bureaucratic headaches of first-order representations (e.g., capture-avoiding substitution).

Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. For example, intersection types distinguish between call-by-name and call-by-value functions, because the subtyping law (A→B)∩(A→C)≤A→(B∩C) is unsound for the latter in the presence of effects. In this paper we develop a proof-theoretic framework for analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered through linear logic, but we propose a fresh origin in Dummett’s program of justifying the logical laws through alternative verificationist or pragmatist “meaning-theories”, which include a bias towards either introduction or elimination rules. We revisit Dummett’s analysis using the tools of Martin-Löf’s judgmental method, and then show how to extend it to a unified polarized logic, with Girard’s “shift” connectives acting as intermediaries. This logic safely combines intuitionistic and dual intuitionistic reasoning principles, while simultaneously admitting a focusing interpretation for the classical sequent calculus. Then, by applying the Curry–Howard isomorphism to polarized logic, we obtain a single programming language in which evaluation order is reflected at the level of types. Different logical notions correspond directly to natural programming constructs, such as pattern-matching, explicit substitutions, values and call-by-value continuations. We give examples demonstrating the expressiveness of the language and type system, and prove a basic but modular type safety result. We conclude with a brief discussion of extensions to the language with additional effects and types, and sketch the sort of explanation this can provide for operationally sensitive typing phenomena.