Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this paper, we develop a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structure-semantics adjointness result and a double categorical universal property of categories of models are also shown.
The framework by Avery @cite_3 is relative to a well-behaved 2-category (which he calls a ). In the basic setting of @math he identifies algebraic theories with identity-on-objects functor from a certain category @math of arities, calling them . In this case, the relationship to our work may be established by the fact that (putting size issues aside) identity-on-objects functors from @math correspond to monoid objects in @math (with the profunctor composition as the monoidal structure). This way we may understand Avery's framework (with respect to the setting @math ) within ours, although for general setting probably we cannot do so. However we remark that for specific examples of settings treated in @cite_3 , it seems that proto-theories therein can be identified with monoid objects in the category of a suitable variant of profunctors.
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Whereas formal category theory is classically considered within a @math -category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a @math -category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding @math defines @math as the free small cocompletion of @math , in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad @math on a hypervirtual double category @math several hypervirtual double categories @math of @math -algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors @math . The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in @math lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in @math . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by as a metric on the class of persistence modules parametrized over the real line. subsequently extended the definition to categories of functors on a poset, the objects in these categories being regarded as generalized persistence modules'. These metrics typically depend on the choice of a lax semigroup of endomorphisms of the poset. The purpose of the present paper is to develop a more general framework for the notion of interleaving distance using the theory of actegories'. Specifically, we extend the notion of interleaving distance to arbitrary categories equipped with a flow, i.e. a lax monoidal action by the monoid @math . In this way, the class of objects in such a category acquires the structure of a Lawvere metric space. Functors that are colax @math -equivariant yield maps that are @math -Lipschitz. This leads to concise proofs of various known stability results from TDA, by considering appropriate colax @math -equivariant functors. Along the way, we show that several common metrics, including the Hausdorff distance and the @math -norm, can be realized as interleaving distances in this general perspective. There are many category-theoretic notions of algebraic theory, including Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. In order to define models of a proto-theory in a category, we need a way of relating the arities of the proto-theory with the objects of the category. This leads to our second central notion, that of an interpretation of arities, or aritation for short. We show that every aritation gives rise to a semantics functor sending proto-theories to models. In fact this functor always has an adjoint, giving a structure-semantics adjunction. Furthermore, we show that the semantics of proto-theories generalises the classical semantics of many existing notions of algebraic theory. Another aim of this thesis is to find a convenient category of monads in the following sense. Every right adjoint gives rise to a monad on its codomain, and more generally so does any functor that admits a codensity monad. However, not all functors have codensity monads. This means that the semantics functor for monads on a category, viewed as a functor into the category of all functors into the base category, does not have a left adjoint. We seek a generalisation of monads with a semantics functor that does have a left adjoint. This can be accomplished with proto-theories, but at a cost. The classical semantics functor for monads is full and faithful; this is a kind of completeness theorem, and is highly desirable in a notion of algebraic theory. However, the semantics functor for general proto-theories is not full and faithful. Thus we seek a generalisation of monads and their semantics for which the semantics functor is both full and faithful and has a left adjoint. We show that such a generalisation is given, for suitable base categories, by a certain kind of topological proto-theory. Rewriting logic extends to concurrent systems with state changes the body of theory developed within the algebraic semantics approach. It is both a foundational tool and the kernel language of several implementation efforts (Cafe, ELAN, Maude). Tile logic extends (unconditional) rewriting logic since it takes into account state changes with side effects and synchronization. It is especially useful for defining compositional models of computation of reactive systems, coordination languages, mobile calculi, and causal and located concurrent systems. In this paper, the two logics are defined and compared using a recently developed algebraic specification methodology, membership equational logic. Given a theory T, the rewriting logic of T is the free monoidal 2-category, and the tile logic of T is the free monoidal double category, both generated by T. An extended version of monoidal 2-categories, called 2VH-categories, is also defined, able to include in an appropriate sense the structure of monoidal double categories. We show that 2VH-categories correspond to an extended version of rewriting logic, which is able to embed tile logic, and which can be implemented in the basic version of rewriting logic using suitable internal strategies. These strategies can be significantly simpler when the theory is uniform. A uniform theory is provided in the paper for CCS, and it is conjectured that uniform theories exist for most process algebras. Goerss--Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model @math -categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable @math -category (such as that of equivariant spectra or of motivic spectra). We study Kan extensions in three weakenings of the Eilenberg-Moore dou- ble category associated to a double monad, that was introduced by Grandis and Par e. To be precise, given a normal oplax double monad T on a double categoryK, we consider the double categories consisting of pseudo T -algebras, 'weak' vertical T -morphisms, hor- izontal T -morphisms and T -cells, where 'weak' means either 'lax', 'colax' or 'pseudo'. Denoting these double categories by Algw(T ), where w = l, c or ps accordingly, our main result gives, in each of these cases, conditions ensuring that (pointwise) Kan extensions can be lifted along the forgetful double functor Algw(T ) ! K. As an application we recover and generalise a result by Getzler, on the lifting of pointwise left Kan extensions along symmetric monoidal enriched functors. As an application of Getzler's result we prove, in suitable symmetric monoidal categories, the existence of bicommutative Hopf monoids that are freely generated by cocommutative comonoids. Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The @math -model structure on @math Equivariant @math -type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho @math and @math Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and @math -categories The category of orthogonal @math -spectra over @math Model structures for parametrized @math -spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation. A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given a (eventually non-semisimple) k-linear abelian ribbon braided category @math with some finiteness conditions we construct a functor from a central extension of N with the set of labels @math ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surfto k-vector spaces. We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra @math for a functor @math , we define a framework for deriving pseudometrics on @math which measure the behavioral distance of states. A crucial step is the lifting of the functor @math on @math to a functor @math on the category @math of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. If @math has a final coalgebra, every lifting @math yields in a canonical way a behavioral distance which is usually branching-time, i.e., it generalizes bisimilarity. In order to model linear-time metrics (generalizing trace equivalences), we show sufficient conditions for lifting distributive laws and monads. These results enable us to employ the generalized powerset construction. We define a monad on the category of complete metric spaces with short maps, which assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich--Wasserstein distance. It is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite powers, which formalizes the intuition that probability measures are limits of finite samples. The proof relies on a new criterion for when an ordinary left Kan extension of lax monoidal functors is a monoidal Kan extension. This colimit characterization allows for the development of integration theory and other things, such as the treatment of measures on spaces of measures, completely without measure theory. We also show that the category of algebras of the Kantorovich monad is equivalent to the category of closed convex subsets of Banach spaces with short affine maps as the morphisms. We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset monad. We exploit the self-duality of the category Rel, i.e., the Kleisli category for the powerset monad, and the category theoretic structures on it that allow us to build models of the untyped λ-calculus, yielding a variant of the Engeler model. We replace the monad for commutative monoids by that for idempotent commutative monoids, which, on Set, is the finite powerset monad. This does not quite yield a distributive law, so requires a little more subtlety, but, subject to that subtlety, it yields exactly the original Engeler construction. An algebraic left Kan extension is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say that the categories in question are universal examples of some categorical structure possessing some kind of internal structure, and so fall within the theory of internal algebra classifiers described in earlier work of the author. In this article conditions of a monad-theoretic nature are identified which give rise to morphisms between such universal objects, which satisfy the key condition of Guitart-exactness, which guarantees the algebraicness of left Kan extending along them. The resulting setting explains the algebraicness of the left Kan extensions arising in operad theory, for instance from the theory of Feynman categories of Kaufmann and Ward, generalisations thereof, and also includes the situations considered by Batanin and Berger in their work on the homotopy theory of algebras of polynomial monads. Applicative functors define an interface to computation that is more general, and correspondingly weaker, than that of monads. First used in parser libraries, they are now seeing a wide range of applications. This paper sets out to explore the space of non-monadic applicative functors useful in programming. We work with a generalization, lax monoidal functors, and consider several methods of constructing useful functors of this type, just as transformers are used to construct computational monads. For example, coends, familiar to functional programmers as existential types, yield a range of useful applicative functors, including left Kan extensions. Other constructions are final fixed points, a limited sum construction, and a generalization of the semi-direct product of monoids. Implementations in Haskell are included where possible. Author(s): Mazel-Gee, Aaron | Advisor(s): Teichner, Peter | Abstract: We develop a theory of model ∞-categories -- that is, of model structures on ∞-categories -- which provides a robust theory of resolutions entirely native to the ∞-categorical context. Using model ∞-categories, we generalize Goerss--Hopkins obstruction theory from spectra to an arbitrary (presentably symmetric monoidal stable) ∞-category. We give a sample application of this generalized obstruction theory in the setting of motivic homotopy theory, where we construct E_∞ structures on the motivic Morava E-theories and compute their automorphism spaces (as E_∞ algebras). We give a general category theoretic formulation of the substitution structure underlying the category theoretic study of variable binding proposed by Fiore, Plotkin, and Turi. This general formulation provides the foundation for their work on variable binding, as well as Tanaka's linear variable binding and variable binding for other binders and for mixtures of binders as for instance in the Logic of Bunched Implications. The key structure developed by was a substitution monoidal structure, from which their formulation of binding was derived; so we give an abstract formulation of a substitution monoidal structure, then, at that level of generality, derive the various category theoretic structures they considered. The central construction we use is that of a pseudo-distributive law between 2-monads on Cat, which suffices to induce a pseudo-monad on Cat, and hence a substitution monoidal structure on the free object on 1. We routinely generalise that construction to account for types. In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually used: deducibility and indistinguishability. Only few results have been obtained (in an ad-hoc way) for equational theories with associative and commutative properties, especially in the case of static equivalence. The main contribution of this paper is to propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of these theories. Our setting relies on the correspondence between a monoidal theory E and a semiring SE which allows us to give an algebraic characterization of the deducibility and indistinguishability problems. As a consequence we recover easily existing decidability results and obtain several new ones. We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between soft' and hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors. In this paper we construct new categorical models for the identity types of Martin-Lof type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren [2009], which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorization system in the sense of Bousfield--Quillen. It turns out that this is not quite enough for a sound model, due to some subtle coherence issues concerned with stability under substitution; and so our first task is to introduce a slightly richer structure, which we call a homotopy-theoretic model of identity types, and to prove that this is sufficient for a sound interpretation. Now, although both Top and SSet are categories endowed with a weak factorization system---and indeed, an entire Quillen model structure---exhibiting the additional structure required for a homotopy-theoretic model is quite hard to do. However, the categories we are interested in share a number of common features, and abstracting these leads us to introduce the notion of a path object category. This is a relatively simple axiomatic framework, which is nonetheless sufficiently strong to allow the construction of homotopy-theoretic models. Now by exhibiting suitable path object structures on Top and SSet, we endow those categories with the structure of a homotopy-theoretic model and, in this way, obtain the desired topological and simplicial models of identity types. Abstract Within the Geometry of Interaction (GoI) paradigm, we present a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation realization of the traced monoidal categories of finite dimensional vector spaces with tensor product as monoidal structure and of finite sets and relations with Cartesian product as monoidal structure, both of them providing a so-called wave-style GoI. The developments in this paper reveal that envisioning state update due to quantum measurement as a process provides a powerful tool for developing high-level approaches to quantum information processing. Sabadini, Walters and others have developed a categorical, machine based theory of concurrency in which there are four essential aspects: a distributive category of data-types; a bicategory Mach whose objects are data types, and whose arrows are input-output machines built from data types; a semantic category (or categories) Sem, suitable to contain the behaviors of machines, and a functor, “behavior”: Mach→Sem. Suitable operations on machines and semantics are found so that the behavior functor preserves these operations. Then, if each machine is decomposable into primitive machines using these operations, the behavior of a general machine is deducible from the behavior of its parts. The theory of non-deterministic finite state automata provides an example of the paradigm and also throws some light on the classical theory of finite state automata.