In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the `morphisms between 0-cells', such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change `nonalgebraically', using categorical fibrations. The resulting `framed bicategories' assemble into 2-categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a `monoidal fibration', meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.
Deep Dive into Framed bicategories and monoidal fibrations.
We begin with the observation that there are really two sorts of bicategories (or 2-categories). This fact is well appreciated in 2-categorical circles, but not as widely known as it ought to be. (In fact, there are other sorts of bicategory, but we will only be concerned with two.)
The first sort is exemplified by the 2-category Cat of categories, functors, and natural transformations. Here, the 0-cells are ‘objects’, the 1-cells are maps between them, and the 2-cells are ‘maps between maps.’ This sort of bicategory is welldescribed by the slogan “a bicategory is a category enriched over categories.”
The second sort is exemplified by the bicategory Mod of rings, bimodules, and bimodule homomorphisms. Here, the 1-cells are themselves ‘objects’, the 2-cells are maps between them, and the 0-cells are a different sort of ‘object’ which play a ‘bookkeeping’ role in organizing the relationships between the 1-cells. This sort of bicategory is well-described by the slogan “a bicategory is a monoidal category with many objects.”
Many notions in bicategory theory work as well for one sort as for the other. For example, the notion of 2-functor (including lax 2-functors as well as pseudo ones) is well-suited to describe morphisms of either sort of bicategory. Other notions, such as that of internal adjunction (or ‘dual pair’), are useful in both situations, but their meaning in the two cases is very different.
However, some bicategorical ideas make more sense for one sort of bicategory than for the other, and frequently it is the second sort that gets slighted. A prime example is the notion of equivalence of 0-cells in a bicategory. This specializes in Cat to equivalence of categories, which is unquestionably the fundamental notion of ‘sameness’ for categories. But in Mod it specializes to Morita equivalence of rings, which, while very interesting, is not the most fundamental sort of ‘sameness’ for rings; isomorphism is. This may not seem like such a big deal, since if we want to talk about when two rings are isomorphic, we can use the category of rings instead of the bicategory Mod. However, it becomes more acute when we consider the notion of biequivalence of bicategories, which involves pseudo 2-functors F and G, and equivalences X ≃ GF X and Y ≃ F GY . This is fine for Cat -like bicategories, but for Mod -like bicategories, the right notion of equivalence ought to include something corresponding to ring isomorphisms instead. This problem arose in [MS06,19.3.5], where two Mod-like bicategories were clearly ’equivalent’, yet the language did not exist to describe what sort of equivalence was meant.
Similar problems arise in many other situations, such as the following.
(i) Cat is a monoidal bicategory in the usual sense, which entails (among other things) natural equivalences (C × D) × E ≃ C × (D × E). But although Mod is ‘morally monoidal’ under tensor product of rings, the associativity constraint is really a ring isomorphism (R ⊗ S) ⊗ T ∼ = R ⊗ (S ⊗ T ), not an invertible bimodule (although it can be made into one). (ii) For Cat -like bicategories, the notions of pseudonatural transformation and modification, making bicategories into a tricategory, are natural and useful. But for Mod -like bicategories, it is significantly less clear what the right sort of higher morphisms are. (iii) The notion of ‘biadjunction’ is well-suited to adjunctions between Catlike bicategories, but fails badly for Mod -like bicategories. Attempts to solve this problem have resulted in some work, such as [Ver92, CKW91, CKVW98], which is closely related to ours.
These problems all stem from essentially the same source: the bicategory structure does not include the correct ‘maps between 0-cells’, since the 1-cells of the bicategory are being used for something else. In this paper, we show how to use an abstract structure to deal with this sort of situation by incorporating the maps of 0-cells separately from the 1-cells. This structure forms a pseudo double category with extra properties, which we call a framed bicategory.
The first part of this paper is devoted to framed bicategories. In § §2-5 we review basic notions about double categories and fibrations, define framed bicategories, and prove some basic facts about them. Then in § §6-10 we apply framed bicategories to resolve the problems mentioned above. We define lax, oplax, and strong framed functors and framed transformations, and thereby obtain three 2-categories of framed bicategories. We then apply general 2-category theory to obtain useful notions of framed equivalence, framed adjunction, and monoidal framed bicategory.
The second part of the paper, consisting of § §11-17, deals with two important ways of constructing framed bicategories. The first, which we describe in §11, starts with a framed bicategory D and constructs a new framed bicategory Mod(D) of monoids and modules in D. The second starts with a different ‘parametrized monoidal structure’ called a monoidal fi
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