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We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.

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Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched call-byvalue calculus. From the perspective of categorical models, the enriched call-by-value calculus suggests a refinement of the traditional Kleisli models of effectful call-by-value languages. The new models can be understood as enriched adjunctions. 1

To Mamuka Jibladze on occasion of his 50th birthday Abstract. We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.

Philosophy of quantum groups as Hopf algebras and their actions and coactions on algebras and graded algebras captures only affine and some projective phenomena in noncommutative algebraic geometry. Such actions have to be rephrased in terms of categories of modules or quasicoherent sheaves, to enable the flexibility in a more general setup of noncommutative locally affine spaces. We sketch here very basic machinery to talk about noncommutative G-spaces, 1- and 2-equivariant objects, noncommutative fiber bundles and quotients. Some of the notions studied here may be useful in approaching categorified constructions in commutative geometry, e.g. associated

Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodule-monad and a algebra-comonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler’s result (to the setting of Hopf Galois extensions).