## FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS (2009)

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@MISC{SHULMAN09framedbicategories,

author = {MICHAEL SHULMAN},

title = {FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS},

year = {2009}

}

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### Abstract

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

### Citations

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Citation Context ... over V , or V -categories. Its vertical arrows are V -functors, its 1-cells are V -distributors, and its 2-cells are V -natural transformations. (Good references for enriched category theory include =-=[Kel82]-=- and [Dub70].) A V -distributor H : B �→ A is simply a V -functor H : A op ⊗ B → V . When A and B have one object, they are just monoids in V , and a distributor between them is a bimodule in V ; thus... |

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Citation Context ...FIBRATIONS 655 such that L(a), R(a), L(l), R(l), L(r), and R(r) are all identities, and such that the standard coherence axioms for a monoidal category or bicategory (such as Mac Lane’s pentagon; see =-=[ML98]-=-) are satisfied. We can think of a double category as an internal category in Cat which is suitably weakened, although this is not strictly true because Cat contains only small categories while we all... |

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Citation Context ...fice. Since we are not writing primarily for category theorists, we have attempted to avoid or explain the more esoteric categorical concepts which arise. A classic reference for 2-category theory is =-=[KS74]-=-; a more modern and comprehensive one (going far beyond what we will need) is [Lac07]. The second important theme of this paper is the mixture of ‘algebraic’ and ‘nonalgebraic’ structures. A monoidal ... |

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Citation Context ...der to get a handle on internal or enriched functors purely bicategorically. In the next appendix we introduce a structure called an equipment which is sometimes used for this purpose, for example in =-=[LS02]-=-.sC. Equipments FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS 733 For the theory of equipments we refer the reader to [Woo82, Woo85, CJSV94, Ver92]. From our point of view, it is natural to introduce th... |

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Citation Context ... V -categories. Its vertical arrows are V -functors, its 1-cells are V -distributors, and its 2-cells are V -natural transformations. (Good references for enriched category theory include [Kel82] and =-=[Dub70]-=-.) A V -distributor H : B �→ A is simply a V -functor H : A op ⊗ B → V . When A and B have one object, they are just monoids in V , and a distributor between them is a bimodule in V ; thus we have an ... |

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Citation Context ...framed adjunction F ⊣ G, the left adjoint F is always a strong framed functor. Sketch of Proof. This actually follows formally from a general 2-categorical result known as ‘doctrinal adjunction’; see =-=[Kel74]-=-. For the non-2-categorically inclined reader we sketch a more concrete version of the proof. We first show that the following composite is an inverse to F⊙ : F M ⊙ F N → F (M ⊙ N): F (M ⊙ N) F (η⊙η) ... |

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Citation Context ... relationship to various parts of bicategory theory. ∼= ∼= (75)sFRAMED BICATEGORIES AND MONOIDAL FIBRATIONS 729 For further detail on connection pairs, we refer the reader to [BS76, BM99] and also to =-=[Fio06]-=-, which proved that connection pairs are equivalent to ‘foldings’. Our presentation of the theory differs from the usual one because we focus on the pseudo case, which turns out to simplify the defini... |

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Citation Context ...l category, considered as a one-object bicategory, an adjunction in C is better known as a dual pair in C , and one speaks of an object Y as being left or right dual to an object X; see, for example, =-=[May01]-=-. When C is symmetric monoidal, left duals and right duals coincide. 5.1. Examples. When C = ModR for a commutative ring R, the dualizable objects are the finitely generated projectives. When C is the... |

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Citation Context ...ver, we would need morphisms and especially transformations between equipments, and it is not immediately obvious how to define these. The approach to constructing a 2-category of equipments taken in =-=[Ver92]-=- is essentially to first make equipments into double categories, as we have done, and define morphisms and transformations of equipments to be morphisms between the corresponding double categories. Th... |

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Citation Context ...es in D. The second starts with a different ‘parametrized monoidal structure’ called a monoidal fibration, and is essentially the same as the construction of the bicategory of parametrized spectra in =-=[MS06]-=-. In §§12–13 we introduce monoidal fibrations, and in §14 we explain the connection to framed bicategories. Then in §15, we combine these two constructions and thereby obtain a natural theory of ‘cate... |

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Citation Context ...tegory theory can be regarded as implicitly working with the framed bicategory Adj(B); frequently 1-cells with right adjoints are called maps and take on a special role. See, for example, [Str81] and =-=[CKW87]-=-. This purely bicategorical approach works well in bicategories like Dist(V ), because, as we mentioned in Example 5.6(iii), the mild condition of ‘Cauchy completeness’ on the V -categories involved i... |

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(Show Context)
Citation Context ...of pure bicategory theory can be regarded as implicitly working with the framed bicategory Adj(B); frequently 1-cells with right adjoints are called maps and take on a special role. See, for example, =-=[Str81]-=- and [CKW87]. This purely bicategorical approach works well in bicategories like Dist(V ), because, as we mentioned in Example 5.6(iii), the mild condition of ‘Cauchy completeness’ on the V -categorie... |

11 |
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Citation Context ...hendieck and his school; see, for example [sga03, Exposé VI]. Modern references include [Joh02a, B1.3] and [Bor94, Ch. 8]. More abstract versions can be found in the 2-categorical literature, such as =-=[Str80]-=-. 3.1. Definition. Let Φ : A → B be a functor, let f : A → C be an arrow in B, and let M be an object of A with Φ(M) = C. An arrow φ : f ∗ M → M in A is cartesian over f if, firstly, Φ(φ) = f: f ∗M φ ... |

9 | The periodic table of ncategories for low dimensions I: degenerate categories and degenerate bicategories
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Citation Context ...d 2-functors between such bicategories do also correspond to monoidal functors, but most transformations between such 2-functors do not give rise to anything resembling a monoidal transformation; see =-=[CG06]-=-. Thus, framed bicategories are a better generalization of monoidal categories than ordinary bicategories are. 6.10. Example. Let C and D be monoidal categories with coequalizers preserved by ⊗, and l... |

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Citation Context ... the right adjoint of a strong monoidal functor is always lax monoidal. These are consequences of ‘doctrinal adjunction’ (see Proposition 8.2 and [Kel74]) and a property called ‘lax-idempotence’ (see =-=[KL97]-=-). In many cases, F0 and G0 are the identity, and the entire adjunction is ‘over B’ in the sense introduced in Definition 12.5. 14.14. Example. Let C have finite limits and finite colimits preserved b... |

6 | 2002) Categories enriched on two sides - Kelly, Labella, et al. |

5 | The periodic table of n-categories for low dimensions ii: degenerate tricategories - Cheng, Gurski |

4 |
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Citation Context ...ual. In contrast to these well-behaved cases, a framed adjunction F : D ⇄ E : G does not generally give rise to a biadjunction D ⇄ E. It does, however, give rise to a local adjunction in the sense of =-=[BP88]-=-; this consists of an oplax 2-functor F : D → E, a lax 2-functor G: E → D, and an adjunction D(A, GB) ⇄ E(F A, B). (78) In a biadjunction, F and G would be pseudo 2-functors and (78) would be an equiv... |

4 |
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Citation Context ...adjoint, and hence by Proposition 8.2 is strong. As is the case for categories, we can also characterize framed adjunctions using universal arrows. A similar result for double categories was given in =-=[Gar07]-=-. Recall that given a functor G: E → D, a universal arrow to G is an arrow η : A → GF A in D, for some object F A ∈ E , such that any other arrow A → GY factors through η via a unique map F A → Y in E... |

4 | Abstract proarrows - Wood - 1982 |

3 |
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Citation Context ... universal property, and the definition merely stipulates that an object satisfying that property exists, rather than choosing a particular such object as part of the structure. In the terminology of =-=[Mak01]-=-, they are virtual operations. The ‘algebraic’ notion corresponding to a fibration Φ: A → B is a pseudofunctor P : B op → Cat. Given a fibration Φ, if we choose a cleavage, then we obtain, for eachsFR... |

3 |
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Citation Context ...rk came from the bicategorical ‘shadows’ of [Pon07], and a desire to explain in what way they are actually the same as the horizontal composition in the bicategory; we will do this in the forthcoming =-=[PS08]-=-. I would like to thank my advisor, Peter May, as well as Kate Ponto, for many useful discussions about these structures; Tom Fiore, for the idea of using double categories; and Joachim Kock and Steph... |

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Citation Context ...o avoid or explain the more esoteric categorical concepts which arise. A classic reference for 2-category theory is [KS74]; a more modern and comprehensive one (going far beyond what we will need) is =-=[Lac07]-=-. The second important theme of this paper is the mixture of ‘algebraic’ and ‘nonalgebraic’ structures. A monoidal category is an algebraic structure: the product is a specified operation on objects. ... |

2 | Fixed point theory and trace for bicategories
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Citation Context ...nal category in Top with a disjoint section adjoined. Certain monoids in Sp arising in this way from the topologized fundamental groupoid ΠM or path-groupoid PM of a space M play an important role in =-=[Pon07]-=-. A good case can be made (see [MS06]) that a monoid in Sp is the right parametrized analogue of a classical ring spectrum, since when its space of objects is a point, it reduces to an orthogonal ring... |

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Citation Context ...sly to Example 15.6, that monoids in Ab(E ) give a good notion of a ‘bundle of rings’. The theory of such relative enriched categories appears to be fairly unexplored; the only references we know are =-=[GG76]-=- and [Prz07]. We will explore this theory more extensively in a later paper; in many ways, it is very similar to classical enriched category theory. We end with one further example of this phenomenon.... |

1 |
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Citation Context ...ple 15.6, that monoids in Ab(E ) give a good notion of a ‘bundle of rings’. The theory of such relative enriched categories appears to be fairly unexplored; the only references we know are [GG76] and =-=[Prz07]-=-. We will explore this theory more extensively in a later paper; in many ways, it is very similar to classical enriched category theory. We end with one further example of this phenomenon. If V is an ... |

1 | Algebraic Tricategories - Gurski - 2006 |

1 | The periodic table in low dimensions i: degenerate categories and degenerate bicategories. To appear in proceedings of the Streetfest. Available online at http://math.unice.fr/∼eugenia/degeneracy - Cheng, Gurski - 2006 |