Recent work of Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a valued field k can be modelled as geometry relative to the quasi-abelian category of Banach spaces, or rather its completion Ind(Ban_k). In this thesis we develop a robust theory of homotopical algebra in Ch(E) for E any sufficiently 'nice' quasi-abelian, or even exact, category. Firstly we provide sufficient conditions on weakly idempotent complete exact categories E such that various categories of chain complexes in E are equipped with projective model structures. In particular we show that as soon as E has enough projectives, the category Ch₊(E) of bounded below complexes is equipped with a projective model structure. In the case that E also admits all kernels we show that it is also true of Ch≥0(E), and that a generalisation of the Dold-Kan correspondence holds. Supplementing the existence of kernels with a condition on the existence and exactness of certain direct limit functors guarantees that the category of unbounded chain complexes Ch(E) also admits a projective model structure. When E is monoidal we also examine when these model structures are monoidal. We then develop the homotopy theory of algebras in Ch(E). In particular we show, under very general conditions, that categories of operadic algebras in Ch(E) can be equipped with transferred model structures. Specialising to quasi-abelian categories we prove our main theorem, which is a vast generalisation of Koszul duality. We conclude by defining analytic extensions of the Koszul dual of a Lie algebra in Ind(Ban_k).

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We nd concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in [4] on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.

This thesis consists of three papers about N-complexes and their uses in categorification. N-complexes are generalizations of chain complexes having a differential d satisfying dN = 0 rather than d2 = 0. Categorification is the process of finding a higher category analog of a given mathematical structure. Paper I: We study a set of homology functors indexed by positive integers a and b and their corresponding derived categories. We show that there is an optimal subcategory in the domain of every functor given by N-complexes with N = a + b. Paper II: In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of N-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this in general and present evidence that the axioms of triangulated categories have a simplicial origin. Paper III: Let n be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of n:th cyclotomic integers.

It is widely accepted within the software engineering community that the support for integration is necessary for requirement models. Several methodologies, such as the role-based software development, that have appeared in the literature are relying on some kind of integration. However, current integration techniques and their tools support are insufficient. In this research, we discuss our solution to the problem. More precisely, we present a general integration approach for scenario-based models, particularly for UML Sequence Diagrams, based on the colimit construction known from category theory. In our approach, Sequence Diagrams are represented by SD-graphs, a special kind of typed graphs. The merge algorithm for SD-graphs is an extension of existing merge operations on sets and graphs. On the one hand, the merge algorithm ensures traceability and guarantees key theoretical properties (e.g., “everything is represented and nothing extra is acquired” during the merge). On the other hand, our formalization of Sequence Diagrams as SD-graphs retains the graphical nature of Sequence Diagrams, yet is amenable to algebraic manipulations. Another important property of our process is that our approach is applicable to other kinds of models as long as they can be represented by typed graphs. A prototype Sequence Diagram integration tool following the approach has been implemented. The tool is not only a fully functional integration tool, but also served as a test bed for our theory and provided feedback for our theoretical framework. To support the discovery and specification of model relationships, we also present a list of high-level merge patterns in this dissertation. We believe our theory and tool are beneficial to both academia and industry, as the initial evaluation has shown that the ideas presented in this dissertation represent promising steps towards the more rigorous management of requirement models. We also present an approach connecting model transformation with source transformation and allowing an existing source transformation language (TXL) to be used for model transformation. Our approach leverages grammar generators to ease the task of creating model transformations and inherits many of the strengths of the underlying transformation language (e.g., efficiency and maturity). / Thesis (Ph.D, Computing) -- Queen's University, 2009-08-28 13:03:08.607

A Kahler category axiomatizes the algebraic geometric theory of Kahler Differentials in an abstract categorical setting. To facilitate this, a Kahler category is equipped with an algebra modality, which endows each object in the image of a specified monad with an associative algebra structure; universal derivations are then required to exist naturally for each of these objects. Moreover, it can be demonstrated that for each T-algebra of said monad there is a natural associative algebra structure. In this paper I will show that under certain conditions on the Kahler category, the universal derivations for the algebras arising from T-algebras exist and arise via a coequalizer. Furthermore, this result is extended to provide an alternative construction for universal derivations for a more general class of algebras, including all algebras in a Kahler category. A prospective categorical formulation of the theory of noncommutative Kahler differentials is then given, and the above said results are shown to apply in this context. Finally, another class of algebras is constructed via a colimit, and the modules of differential forms for these algebras is computed.

Γίνεται αναφορά στην Ιστορία της έννοιας του καθολικού, του συγκεκριμένου καθολικού, και της Θεωρίας Κατηγοριών. Κατόπιν, παρουσιάζεται η απόπειρα τυποποίησης μέσω της Θεωρίας των Κατηγοριών εννοιών με το οντοτολογικό status του συγκεκριμένου καθολικού, όπως και η χρήση της έννοιας του συγκεκριμένου καθολικού ως ερμηνείας κάποιων καθολικών κατασκευών που απαντώνται στη Θεωρία Κατηγοριών. / Category theory as mathematical theory of concrete universals and concrete universality as an interpretation of universal objects in special categories.

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion. To begin, we turn to type theory. We construct a modal λ-calculus, called Intensional PCF, which supports non-functional operations at modal types. Moreover, by adding Löb's rule from provability logic to the calculus, we obtain a type-theoretic interpretation of intensional recursion. The combination of these two features is shown to be consistent through a confluence argument. Following that, we begin searching for a semantics for Intensional PCF. We argue that 1-category theory is not sufficient, and propose the use of P-categories instead. On top of this setting we introduce exposures, which are P-categorical structures that function as abstractions of well-behaved intensional devices. We produce three examples of these structures, based on Gödel numberings on Peano arithmetic, realizability theory, and homological algebra. The language of exposures leads us to a P-categorical analysis of intensional recursion, through the notion of intensional fixed points. This, in turn, leads to abstract analogues of classic intensional results in logic and computability, such as Gödel's Incompleteness Theorem, Tarski's Undefinability Theorem, and Rice's Theorem. We are thus led to the conclusion that exposures are a useful framework, which we propose as a solid basis for a theory of intensionality. In the final chapters of the thesis we employ exposures to endow Intensional PCF with an appropriate semantics. It transpires that, when interpreted in the P-category of assemblies on the PCA K1, the Löb rule can be interpreted as the type of Kleene's Second Recursion Theorem.

This thesis presents a characterization of those categories with weak factorization systems that can interpret the theory of intensional dependent type theory with Σ, Π, and identity types. We use display map categories to serve as models of intensional dependent type theory. If a display map category (C, D) models Σ and identity types, then this structure generates a weak factorization system (L, R). Moreover, we show that if the underlying category C is Cauchy complete, then (C, R) is also a display map category modeling Σ and identity types (as well as Π types if (C, D) models Π types). Thus, our main result is to characterize display map categories (C, R) which model Σ and identity types and where R is part of a weak factorization system (L, R) on the category C. We offer three such characterizations and show that they are all equivalent when C has all finite limits. The first is that the weak factorization system (L, R) has the properties that L is stable under pullback along R and all maps to a terminal object are in R. We call such weak factorization systems type theoretic. The second is that the weak factorization system has what we call an Id-presentation: it can be built from certain categorical structure in the same way that a model of Σ and identity types generates a weak factorization system. The third is that the weak factorization system (L, R) is generated by a Moore relation system. This is a technical tool used to establish the equivalence between the first and second characterizations described. To conclude the thesis, we describe a certain class of convenient categories of topological spaces (a generalization of compactly generated weak Hausdorff spaces). We then construct a Moore relation system within these categories (and also within the topological topos) and thus show that these form display map categories with Σ and identity types (as well as Π types in the topological topos).

The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.

A Kahler category axiomatizes the algebraic geometric theory of Kahler Differentials in an abstract categorical setting. To facilitate this, a Kahler category is equipped with an algebra modality, which endows each object in the image of a specified monad with an associative algebra structure; universal derivations are then required to exist naturally for each of these objects. Moreover, it can be demonstrated that for each T-algebra of said monad there is a natural associative algebra structure. In this paper I will show that under certain conditions on the Kahler category, the universal derivations for the algebras arising from T-algebras exist and arise via a coequalizer. Furthermore, this result is extended to provide an alternative construction for universal derivations for a more general class of algebras, including all algebras in a Kahler category. A prospective categorical formulation of the theory of noncommutative Kahler differentials is then given, and the above said results are shown to apply in this context. Finally, another class of algebras is constructed via a colimit, and the modules of differential forms for these algebras is computed.