mac lane categories for the working mathematician pdf

Moreover, the limit-colimit coincidence from domain theory can be generalized to the unenriched setting and we show that, under suitable assumptions, a wide class of endofunctors has canonical fixed points. As a special case, we show how to define biproducts up to isomorphism in an arbitrary category without assuming any enrichment. We formalize the concept of an ambilimit, and show that it captures known cases. Using cofree dagger categories, the theory of dagger limits can be leveraged to provide an enrichment-free understanding of limit-colimit coincidences in ordinary category theory. We develop a notion of a dagger limit that we show is suitable in the following ways: it subsumes special cases known from the literature dagger limits are unique up to unitary isomorphism a wide class of dagger limits can be built from a small selection of them dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor dagger limits can be built from ordinary limits in the presence of polar decomposition dagger limits commute with dagger colimits in many cases. basic category theory adapts to dagger categories. In this thesis we study the dagger in its own right and show how. Dagger categories with additional structure have been studied under different names in categorical quantum mechanics, algebraic field theory and homological algebra, amongst others. a contravariant involutive identity-on-objects endofunctor. All Concepts are Kan Extensions.- Table of Terminology.Ī dagger category is a category equipped with a functorial way of reversing morphisms, i.e.

$mac lane categories for the working mathematician pdf$

Categories, Functors and Natural Transformations.- 1.

Mac lane categories for the working mathematician pdf