A double-dimensional approach to formal category theory

Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual 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 an augmented virtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in an augmented virtual double category, and compared to that of a good yoneda structure on a $2$-category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding $y \colon A \to \hat A$ defines $\hat A$ as the free small cocompletion of $A$, in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad $T$ on an augmented virtual double category $\mathcal K$ several augmented virtual double categories $T\text-\mathsf{Alg}_{(v, w)}$ of $T$-algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors $T\text-\mathsf{Alg}_{(v, w)} \to \mathcal K$. The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in $\mathcal K$ lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in $T\text-\mathsf{Alg}_{(v, w)}$. 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".Comment: Draft. A streamlined and expanded version of Sections 1, 2 and 3 is available as arXiv:1910.11189. v2: main notion 'hypervirtual double category' has been renamed as 'augmented virtual double category'; several new results; some corrections. Comments are welcom

On pointwise Kan extensions in double categories

In this paper we consider a notion of pointwise Kan extension in double categories that naturally generalises Dubuc's notion of pointwise Kan extension along enriched functors. We show that, when considered in equipments that admit opcartesian tabulations, it generalises Street's notion of pointwise Kan extension in 2-categories.Comment: 38 pages. In v2 the proofs of Propositions 4.2 and 4.4 have been simplified; this is the final version, as it appears in Theory and Applications of Categories, Vol. 29, 201

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories. This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for publication in JPA

Augmented virtual double categories

In this article the notion of virtual double category (also known as fc-multicategory) is extended as follows. While cells in a virtual double category classically have a horizontal multi-source and single horizontal target, the notion of augmented virtual double category introduced here extends the latter notion by including cells with empty horizontal target as well. Any augmented virtual double category comes with a built-in notion of "locally small object" and we describe advantages of using augmented virtual double categories as a setting for formal category rather than 2-categories, which are classically equipped with a notion of "admissible object" by means of a yoneda structure in the sense of Street and Walters. An object is locally small precisely if it admits a horizontal unit, and we show that the notions of augmented virtual double category and virtual double category coincide in the presence of all horizontal units. Without assuming the existence of horizontal units we show that most of the basic theory for virtual double categories, such as that of restriction and composition of horizontal morphisms, extends to augmented virtual double categories. We introduce and study in augmented virtual double categories the notion of "pointwise" composition of horizontal morphisms, which formalises the classical composition of profunctors given by the coend formula.Comment: This article comprises a streamlined and expanded version of Sections 1, 2 and 3 of arXiv:1511.04070. v2 contains several improvements following the referee's suggestions. This is the final version as published in TA

Degrading lists

Post-print (lokagerð höfundar)We discuss the relationship between monads and their known generalisation, graded monads, which are especially useful for modelling computational effects equipped with a form of sequential composition. Specifically, we ask if a graded monad can be extended to a monad, and when such a degrading is in some sense canonical. Our particular examples are the graded monads of lists and non-empty lists indexed by their lengths, which gives us a pretext to study the space of all (non-graded) monad structures on the list and non-empty list endofunctors. We show that, in both cases, there exist infinitely many monad structures. However, while there are at least two ways to complete the graded monad structure on length-indexed lists to a monad structure on the list endofunctor, such a completion for non-empty lists is unique.This research was supported by the Icelandic Research Fund project grant no. 196323-052. T.U. was also supported by the Estonian Ministry of Education and Research institutional research grant no. IUT33-13.Peer reviewed (ritrýnd grein

Formal category theory in augmented virtual double categories

Abridged abstract: In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the notions of Kan extension and Yoneda embedding $y_A \colon A \to \hat A$. The latter includes a formal notion of presheaf object $\hat A$ which recovers, for instance, the classical notions of enriched category of enriched presheaves, enriched category of small enriched presheaves, and power object in a finitely complete category, as well as the notion of Vietoris space of downward-closed subsets of a closed-ordered closure space. We show that the Yoneda embeddings of the Yoneda structure associated to a 2-topos, as constructed by Weber, are instances of our formal notion too. We generalise to monoidal augmented virtual double categories $\mathcal K$ the following fact for finitely complete categories $\mathcal E$ with subobject classifier $\Omega$: $\mathcal E$ has power objects if and only if $\Omega$ is exponentiable. More precisely, given a Yoneda embedding $y_I \colon I \to \hat I$ for the monoidal unit $I$ of $\mathcal K$ and given any `unital' object $A$ in $\mathcal K$, we prove that $y_A \colon A \to \hat A$ exists if and only if the inner hom $[A^\circ, \hat I]$ does, with $A^\circ$ the `horizontal dual' of $A$, and in that case $\hat A \cong [A^\circ, \hat I]$. We end by formalising the classical notions of exact square, total category and `small' cocompletion; the latter in an appropriate sense. Throughout we compare our formalisations to their corresponding 2-categorical counterparts. Our approach has several advantages. For example the structure of augmented virtual double categories naturally allows us to isolate conditions that ensure small cocompleteness of formal presheaf objects $\hat A$.Comment: 97 pages. Comprises a streamlined and much expanded version of Sections 4 and 5 of the draft paper arXiv:1511.0407