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Comparing composites of left and right derived functors
 In preparation
"... Abstract. We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and rig ..."
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Abstract. We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions in 2categories, then gives us canonical ways to compare composites of left and right derived functors. Contents
A Categorical Treatment of Ornaments
"... Abstract—Ornaments aim at taming the multiplication of specialpurpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these specialpurpose datatypes through their common ..."
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Abstract—Ornaments aim at taming the multiplication of specialpurpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these specialpurpose datatypes through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, we gain a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts.
Abstract. KAN EXTENSIONS IN DOUBLE CATEGORIES (ON WEAK DOUBLE CATEGORIES, PART III)
"... are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category. ..."
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are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category.
EXPONENTIABILITY VIA DOUBLE CATEGORIES
"... Cat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case where B is a finite poset. In [23], Street showed that Y � B is exponentiable in Cat/B if and only if the corresponding normal lax functor B � ..."
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Cat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case where B is a finite poset. In [23], Street showed that Y � B is exponentiable in Cat/B if and only if the corresponding normal lax functor B � Cat is a pseudofunctor. Using our generalized equivalence, we show that a morphism Y � B is exponentiable in D0/B if and only if the corresponding normal lax functor B � D is a pseudofunctor plus an additional condition that holds for all X �!B in Cat. Thus, we obtain a single theorem which yields characterizations of certain exponentiable morphisms of small categories, topological spaces, locales, and posets. 1.
SPAN, COSPAN, AND OTHER DOUBLE CATEGORIES
"... Abstract. Given a double category D such that D0 has pushouts, we characterize oplax/lax adjunctions D Cospan(D0) for which the right adjoint is normal and restricts to the identity on D0, where Cospan(D0) is the double category on D0 whose vertical morphisms are cospans. We show that such a pair ex ..."
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Abstract. Given a double category D such that D0 has pushouts, we characterize oplax/lax adjunctions D Cospan(D0) for which the right adjoint is normal and restricts to the identity on D0, where Cospan(D0) is the double category on D0 whose vertical morphisms are cospans. We show that such a pair exists if and only if D has companions, conjoints, and 1cotabulators. The right adjoints are induced by the companions and conjoints, and the left adjoints by the 1cotabulators. The notion of a 1cotabulator is a common generalization of the symmetric algebra of a module and ArtinWraith glueing of toposes, locales, and topological spaces. 1.