Abstract The formal theory of monads shows that much of the theory of monads can be developed in the abstract at the level of 2-categories. This means that results about monads can be established once and for all and simply instantiated in settings such as enriched category theory. Unfortunately, these results can be hard to reason about as they involve more abstract machinery. In this paper, we present the formal theory of monads in terms of string diagrams — a graphical language for 2-categorical calculations. Using this perspective, we show that many aspects of the theory of monads, such as the Eilenberg–Moore and Kleisli resolutions of monads, liftings, and distributive laws, can be understood in terms of systematic graphical calculational reasoning. This paper will serve as an introduction both to the formal theory of monads and to the use of string diagrams, in particular, their application to calculations in monad theory.

We present a categorical theory of the composition methods in finite model theory -- a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the composition methods are Feferman--Vaught--Mostowski (FVM) type theorems, which characterize how logical equivalence behaves under composition and transformation of models. Our results are developed by extending the recently introduced game comonad semantics for model comparison games. This level of abstraction allow us to give conditions yielding FVM type results in a uniform way. Our theorems are parametric in the classes of models, logics and operations involved. Furthermore, they naturally account for the existential and positive existential fragments, and extensions with counting quantifiers of these logics. We also reveal surprising connections between FVM type theorems, and classical concepts in the theory of monads. We illustrate our methods by recovering many classical theorems of practical interest, including a refinement of a previous result by Dawar, Severini, and Zapata concerning the 3-variable counting logic and cospectrality. To highlight the importance of our techniques being parametric in the logic of interest, we prove a family of FVM theorems for products of structures, uniformly in the logic in question, which cannot be done using specific game arguments. This is an extended version of the LiCS 2023 conference paper of the same name. arXiv admin note: substantial text overlap with arXiv:2304.10196

We present a categorical theory of the composition methods in finite model theory – a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the composition methods are Feferman–Vaught–Mostowski (FVM) type theorems, which characterize how logical equivalence be-haves under composition and transformation of models.Our results are developed by extending the recently introduced game comonad semantics for model comparison games. This level of abstraction allow us to give conditions yielding FVM type results in a uniform way. Our theorems are parametric in the classes of models, logics and operations involved. Furthermore, they naturally account for the positive existential fragment, and extensions with counting quantifiers of these logics. We also reveal surprising connections between FVM type theorems, and classical concepts in the theory of monads.We illustrate our methods by recovering many classical theorems of practical interest, including a refinement of a previous result by Dawar, Severini, and Zapata concerning the 3-variable counting logic and cospectrality. To highlight the importance of our techniques being parametric in the logic of interest, we prove a family of FVM theorems for products of structures, uniformly in the logic in question, which cannot be done using specific game arguments.

Game comonads, introduced by Abramsky, Dawar and Wang and developed by Abramsky and Shah, give an interesting categorical semantics to some Spoiler-Duplicator games that are common in finite model theory. In particular they expose connections between one-sided and two-sided games, and parameters such as treewidth and treedepth and corresponding notions of decomposition. In the present paper, we expand the realm of game comonads to logics with generalised quantifiers. In particular, we introduce a comonad graded by two parameters $n \leq k$ such that isomorphisms in the resulting Kleisli category are exactly Duplicator winning strategies in Hella's $n$-bijection game with $k$ pebbles. We define a one-sided version of this game which allows us to provide a categorical semantics for a number of logics with generalised quantifiers. We also give a novel notion of tree decomposition that emerges from the construction.

Monads are commonplace in computer science, and can be composed using Beck's distributive laws. Unfortunately, finding distributive laws can be extremely difficult and error-prone. The literature contains some general principles for constructing distributive laws. However, until now there have been no such techniques for establishing when no distributive law exists. We present three families of theorems for showing when there can be no distributive law between two monads. The first widely generalizes a counterexample attributed to Plotkin. It covers all the previous known no-go results for specific pairs of monads, and includes many new results. The second and third families are entirely novel, encompassing various new practical situations. For example, they negatively resolve the open question of whether the list monad distributes over itself, reveal a previously unobserved error in the literature, and confirm a conjecture made by Beck himself in his first paper on distributive laws. In addition, we establish conditions under which there can be at most one possible distributive law between two monads, proving various known distributive laws to be unique.

In previous work ([1], [2], [3]), it has been shown how a range of model comparison games which play a central role in finite model theory, including Ehrenfeucht-Fraïssé, pebbling, and bisimulation games, can be captured in terms of resource-indexed comonads on the category of relational structures. Moreover, the coalgebras for these comonads capture important combinatorial parameters such as tree-width and tree-depth.The present paper extends this analysis to quantifier-guarded fragments of first-order logic. We give a systematic account, covering atomic, loose and clique guards. In each case, we show that coKleisli morphisms capture winning strategies for Duplicator in the existential guarded bisimulation game, while back-and-forth bisimulation, and hence equivalence in the full guarded fragment, is captured by spans of open morphisms. We study the coalgebras for these comonads, and show that they correspond to guarded tree decompositions. We relate these constructions to a syntax-free setting, with a comonad on the category of hypergraphs.

Monads are commonplace in computer science, and can be composed using Beck's distributive laws. Unfortunately, finding distributive laws can be extremely difficult and error-prone. The literature contains some principles for constructing distributive laws. However, until now there have been no general techniques for establishing when no such law exists. We present two families of theorems for showing when there can be no distributive law for two monads. The first widely generalizes a counterexample attributed to Plotkin. It covers all the previous known no-go results for specific pairs of monads, and includes many new results. The second family is entirely novel, encompassing various new practical situations. For example, it negatively resolves the open question of whether the list monad distributes over itself, and also reveals a previously unobserved error in the literature.

Considering resource usage is a powerful insight in the analysis of many phenomena in the sciences. Much of the current research on these resource theories focuses on the analysis of specific resources such quantum entanglement, purity, randomness or asymmetry. However, the mathematical foundations of resource theories are at a much earlier stage, and there has been no satisfactory account of quantitative aspects such as costs, rates or probabilities. We present a categorical foundation for quantitative resource theories, derived from enriched category theory. Our approach is compositional, with rich algebraic structure facilitating calculations. The resulting theory is parameterized, both in the quantities under consideration, for example costs or probabilities, and in the structural features of the resources such as whether they can be freely copied or deleted. We also achieve a clear separation of concerns between the resource conversions that are freely available, and the costly resources that are typically the object of study. By using an abstract categorical approach, our framework is naturally open to extension. We provide many examples throughout, emphasising the resource theoretic intuitions for each of the mathematical objects under consideration.

Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical systems and network theory. When investigating a new application, the question arises of how to identify a suitable process theoretic model. We present a conceptually motivated parameterized framework for the construction of models for process theories. Our framework generalizes the notion of binary relation along four axes of variation, the truth values, a choice of algebraic structure, the ambient mathematical universe and the choice of proof relevance or provability. The resulting categories are preorder-enriched and provide analogues of relational converse and taking graphs of maps. Our constructions are functorial in the parameter choices, establishing mathematical connections between different application domains. We illustrate our techniques by constructing many existing models from the literature, and new models that open up ground for further development.