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Whereas string diagrams for strict monoidal categories are well understood, and have found application in several fields of Computer Science, graphical formalisms for non-strict monoidal categories are far less studied. In this paper, we provide a presentation by generators and relations of string diagrams for non-strict monoidal categories, and show how this construction can handle applications in domains such as digital circuits and programming languages. We prove the correctness of our construction, which yields a novel proof of Mac Lane's strictness theorem. This in turn leads to an elementary graphical proof of Mac Lane's coherence theorem, and in particular allows for the inductive construction of the canonical isomorphisms in a monoidal category.

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Inspired by the seminal work of Hyland, Plotkin, and Power on the combination of algebraic computational effects via sum and tensor, we develop an analogous theory for the combination of quantitative algebraic effects. Quantitative algebraic effects are monadic computational effects on categories of metric spaces, which, moreover, have an algebraic presentation in the form of quantitative equational theories, a logical framework introduced by Mardare, Panangaden, and Plotkin that generalises equational logic to account for a concept of approximate equality. As our main result, we show that the sum and tensor of two quantitative equational theories correspond to the categorical sum (i.e., coproduct) and tensor, respectively, of their effects qua monads. We further give a theory of quantitative effect transformers based on these two operations, essentially providing quantitative analogues to the following monad transformers due to Moggi: exception, resumption, reader, and writer transformers. Finally, as an application, we provide the first quantitative algebraic axiomatizations to the following coalgebraic structures: Markov processes, labelled Markov processes, Mealy machines, and Markov decision processes, each endowed with their respective bisimilarity metrics. Apart from the intrinsic interest in these axiomatizations, it is pleasing they have been obtained as the composition, via sum and tensor, of simpler quantitative equational theories.

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We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in cryptography. Our model is able to incorporate computational security, set-up assumptions and various attack models such as colluding or independently acting subsets of adversaries in a modular, flexible fashion. We conclude by using string diagrams to rederive the security of the one-time pad, correctness of Diffie-Hellman key exchange and no-go results concerning the limits of bipartite and tripartite cryptography, ruling out e.g., composable commitments and broadcasting. On the way, we exhibit two categorical constructions of resource theories that might be of independent interest: one capturing resources shared among multiple parties and one capturing resource conversions that succeed asymptotically. This is a corrected version of the paper arXiv:2208.13232 published originally on December 18, 2023.

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We show that every continuous valuation on a locally convex, locally convex-compact, sober topological cone $\mathfrak{C}$ has a barycenter. This barycenter is unique, and the barycenter map $\beta$ is continuous, hence is the structure map of a $\mathbf V_{\mathrm w}$-algebra, i.e., an Eilenberg-Moore algebra of the extended valuation monad on the category of $T_0$ topological spaces; it is, in fact, the unique $\mathbf V_{\mathrm w}$-algebra that induces the cone structure on $\mathfrak{C}$.

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We tackle the challenge of ensuring the deadlock-freedom property for message-passing processes that communicate asynchronously in cyclic process networks. Our contributions are twofold. First, we present Asynchronous Priority-based Classical Processes (APCP), a session-typed process framework that supports asynchronous communication, delegation, and recursion in cyclic process networks. Building upon the Curry-Howard correspondences between linear logic and session types, we establish essential meta-theoretical results for APCP, most notably deadlock freedom. Second, we present a new concurrent $\lambda$-calculus with asynchronous session types, dubbed LASTn. We illustrate LASTn by example and establish its meta-theoretical results; in particular, we show how to soundly transfer the deadlock-freedom guarantee from APCP. To this end, we develop a translation of terms in LASTn into processes in APCP that satisfies a strong formulation of operational correspondence.

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