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 hypervirtual double categories, a notion extending that of virtual double category (also known as fc-multicategory) by adding cells with nullary target.This paper starts by introducing the notion of hypervirtual double category, followed by describing its basic theory and its relation to other types of double category. After this the notion of 'weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. The first of these generalises Borceux-Kelly's notion of Kan extension along enriched functors, the second one generalises Street's notion of pointwise Kan extension in 2-categories, and the third is a combination of the other two; these stronger notions are compared. The notion of yoneda embedding is then considered in a hypervirtual 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 : A → A defines 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 a hypervirtual double category K several hypervirtual double categories T -Alg (v,w) of T -algebras are associated, one for each pair of types of weak coherence satisfied by the Talgebras and their morphisms respectively. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors T -Alg (v,w) → K. The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in K lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in T -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". MotivationCentral to classical category theory is the Yoneda lemma which, for a locally small category A, describes the position of the representable presheaves A(-, x) within * Part of this paper was written during a visit to Macquarie University, while its main results formed the subject of a series of talks at the Australian Category Seminar. I would like to thank

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.

In this article the notion of virtual double category [Cruttwell-Shulman, 2010] (also known as (fc-)multicategory [Burroni, 1971;Leinster, 2004]) 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 [Street-Walters, 1978].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.

In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the notions of Kan extension and Yoneda embedding yA : A → A. The latter includes a formal notion of presheaf object 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 K the following fact for finitely complete categories E with subobject classifier Ω: E has power objects if and only if Ω is exponentiable. More precisely, given a Yoneda embedding yI : I → I for the monoidal unit I of K and given any 'unital' object A in K, we prove that yA : A → A exists if and only if the inner hom [A • , I] does, with A • the 'horizontal dual' of A, and in that case A ∼ = [A • , I]. In fact we obtain a more general result that, given a functor F : K → L of augmented virtual double categories and 'universal morphisms' A − − → F A ′ and F P ′ → P in L, allows a Yoneda embedding yA : A → P in L to be "pulled back" to a Yoneda embedding y A ′ : A ′ → P ′ in K.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 A.