A double-dimensional approach to formal category theory Seerp Roald Koudenburg∗ Draft as of October 24, 2019 Notes • The recent paper “Augmented virtual double categories”, avail- able as arxiv:1910.11189, contains a streamlined and ex- panded version of Sections 1, 2 and 3 below; I encourage read- ers of these sections to read the new paper instead. • The first version of this draft (November 2015) used the term “hypervirtual double category” for its main notion, where presently “augmented virtual double category” is used instead. Abstract Whereas formal category theory is classically considered within a 2-cate- gory, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double cat- egories, 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 augmented virtual 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 an augmented virtual double category is considered, together with three strength- enings. The first of these generalises Borceux-Kelly’s notion of Kan extension along enriched functors, the second one generalises Street’s notion of point- wise Kan extension in 2-categories, and the third is a combination of the other arXiv:1511.04070v2 [math.CT] 25 Oct 2019 two; these stronger notions are compared. 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 em- b b bedding 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 an augmented virtual double K T T category several augmented virtual double categories -Alg(v,w) of -al- gebras are associated, one for each pair of types of weak coherence satisfied ∗Part of this paper was written during a visit to Macquarie University, where its main results formed the subject of a series of talks at the Australian Category Seminar; I am grateful to the Macquarie University Research Centre for its funding of my visit. I would like to thank Richard Garner, Mark Weber and Ramón Abud Alcalá for helpful discussions, and Robert Paré for suggesting the term “augmented virtual double category”. 1 by the T -algebras and their morphisms respectively. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful T → K functors -Alg(v,w) . 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 T again define free small cocompletions, now in -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 cocom- pletion, as well as obtaining a “monoidal Yoneda lemma”. Motivation Central 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 the category A of all presheaves Aop → Set. More precisely, in its “parametrised” form, the Yoneda lemma states that the yoneda embedding y: A → A: x 7→ A(–, x) satisfies the followingb property: any profunctor J : A −7−→ B—that is a functor J : Aop × B → Set—induces a functor J λ : B → A equipped with bijectionsb λ ∼ A(y x, J y) = J(x,b y), λ ∼ λ that combine to form an isomorphismb A(y –,J –) = J of profunctors. Indeed, J can be defined as J λy := J(–,y). The main observation motivating thisb paper is that, given a monoidal structure ⊗ on A, the above “lifts” to a “monoidal Yoneda lemma” as follows. Recall that a monoidal structure on A induces such a structure ⊗ on A, that is given by Day convolution [Day70] u,v∈A b b (p ⊗ q)(x) := A(x, u ⊗ v) × pu × qv, where p, q ∈ A, Z and withb respect to which y : A → A forms a pseudomonoidal functor—thatb is, it comes equipped with coherent isomorphisms ¯y: y x ⊗ y y =∼ y(x ⊗ y). Thus a pseudomonoidal functor, (y, ¯y): (A, ⊗b) → (A, ⊗) satisfies the following monoidal variant of the property above: any lax monoidal profunctorb J : A −7−→ B—equipped ¯ with coherent morphisms J : J(x1,y1) × J(x2b,yb2) → J(x1 ⊗ x2,y1 ⊗ y2)—induces a lax monoidal functor J λ : B → A equipped with an isomorphism of lax monoidal profunctors A(y –,J λ –) =∼ J. In detail: we can take J λ to be as defined before, λ λ λ λ equipped with coherence morphismsb J¯ : J y1⊗J y2 ⇒ J (y1⊗y2) that are induced by the compositesb id ×J¯ b A(x, u⊗v)×J(u,y1)×J(v,y2) −−−→ A(x, u⊗v)×J(u⊗v,y1 ⊗y2) → J(x, y1 ⊗y2), where the unlabelled morphism is induced by the functoriality of J in A. The principal aim of this paper is to formalise the way in which the classical Yoneda lemma in the presence of a monoidal structure leads to the monoidal Yoneda lemma, as described above; thus allowing us to (a) generalise the above to other types of algebra, such as double categories and representable topological spaces; (b) characterise lax monoidal profunctors J whose induced lax monoidal functors J λ are pseudomonoidal, and likewise for other types of algebraic morphism; (c) recover classical results in the setting of monoidal categories, such as (y, ¯y) exhibiting (A, ⊗) as the “free monoidal cocompletion” of (A, ⊗) (see [IK86]), and generalise these to other types of algebra. b b 2 Augmented virtual double categories While the classical Yoneda lemma has been formalised in the setting of 2-cate- gories, in the sense of the yoneda structures of Street and Walters [SW78] (see also [Web07]), it is unclear how the monoidal Yoneda lemma, as described above, fits into this framework. Indeed the problem is that it concerns two types of morphism between monoidal categories, both lax monoidal functors and lax monoidal pro- functors, and it is not clear how to arrange both types into a 2-category equipped with a yoneda structure. In particular lax monoidal structures on representable profunctors C(f –, –), where f : A → C is a functor, correspond to colax monoidal structures on f (see Theorem 6.15 below for the generalisation of this to other types of algebra). The natural setting in which to consider two types of morphism is that of a (pseudo-)double category, where one type is regarded as being horizontal and the other as being vertical, while one considers square-shaped cells J A B f φ g C D K between them. Double categories however form a structure that is too strong for some of the types of algebra that we would like to consider, as it requires compo- sitions for both types of morphism while, for example, the composite of “double profunctors” J : A −7−→ B and H : B −7−→ E, between double categories A, B and E, does in general not exist. While the latter is a consequence of, roughly speaking, the algebraic structures of J and H being incompatible in general, composites of profunctor-like morphisms may also fail to exist because of size issues. Indeed, recall from [FS95] that, for a locally small category A, the category of presheaves A need not be locally small. Consequently in describing the classical Yoneda lemma for locally small categories it is, on one hand, necessary to consider categories A,b B, . that might have large hom-sets while, on the other hand, the property satisfied by the yoneda embedding is stated in terms of ‘small’ profunctors—that is small- set-valued functors J : Aop × B → Set—between such categories; in general, such profunctors do not compose either. This classical situation is typical: for a descrip- tion of most of the variations of the Yoneda lemma given in this paper, one considers a double-dimensional setting whose objects A, B, . are “large in size” while the size of the horizontal morphisms J : A −7−→ B between them is “small”. In view of the previous we, instead of double categories, consider a weaker notion as the right notion for our double-dimensional approach to formal category theory. This weaker notion is extends slightly that of ‘virtual double category’ [CS10] which, analogous to the generalisation of monoidal category to ‘multicategory’, does not require a horizontal composition but, instead of the square-shaped cells above, has cells of the form below, with a sequence of horizontal morphisms as horizontal source. Introduced by Burroni [Bur71] under the name ‘fc-catégorie’, where fc denotes the ‘free category’-monad, virtual double categories have also been called ‘fc-multicategories’ [Lei04]. J1 Jn A0 A1 ··· An′ An f φ g C D K Both settings mentioned above can be considered as a virtual double category: there 3 is a virtual double category with (possibly large) categories as objects, functors as vertical morphisms and small profunctors as horizontal morphisms, and likewise one with (possibly large) double categories, ‘double functors’ and ‘small’ double profunctors. These virtual double categories however miss an ingredient crucial to the theory of (double) categories: they do not contain transformations f ⇒ g between (double) functors f and g : A → C into (double) categories C with large hom-sets.