## A double bicategory of cobordisms with corners arXiv:0611930

Citations: | 6 - 0 self |

### BibTeX

@MISC{Morton_adouble,

author = {Jeffrey Morton},

title = {A double bicategory of cobordisms with corners arXiv:0611930},

year = {}

}

### OpenURL

### Abstract

Abstract. Interest in cobordism categories arises in areas from topology to theoretical physics, and in particular in Topological Quantum Field Theories (TQFT’s). These categories have manifolds as objects, and cobordisms between them as morphisms, have- that is, manifolds of one dimension higher whose boundary decomposes into the source and target. Since the boundary of a boundary is empty, this formulation cannot account for cobordisms between manifolds with boundary. This is needed to describe open-closed TQFT’s, and more generally, “extended TQFT’s”. We describe a framework for describing these, in the form of what we call a Verity double bicategory. This is similar to a double category, but with properties holding only up to certain 2-morphisms. We show how a broad general class of examples arise from a construction involving spans (or cospans) in suitable settings, and how this gives cobordisms between cobordisms when we start with the category of manifolds.

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Citation Context ...lize in this paper. Cobordism theory goes back to the work of René Thom [Th], and is closely related to homotopy theory. A good introductory discussion suitable for our needs is found, e.g. in Hirsch =-=[Hi]-=-. There is substantial research on many questions in cobordism theory, such as references. Two manifolds S1, S2 are cobordant if there is a compact manifold with boundary, M, such that ∂M is isomorphi... |

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Citation Context ...ilar to TQFT’s, for which the theory is defined not on cobordisms, but on manifolds with corners. Crane and Yetter [CY], describe the algebraic structure of TQFT’s and extended TQFT’s. Baez and Dolan =-=[BD]-=- summarize the connection between TQFT’s and higher category theory, in the form of the Extended TQFT Hypothesis, suggesting that all extended TQFT’s can be viewed as representations of a certain kind... |

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Citation Context ...sms. 2.1. Presentation. It was shown by Abrams [Ab] that 2Cob can be seen as the free symmetric monoidal category on a Frobenius object. (Another good exposition of this was developed by Joachim Kock =-=[Ko]-=-.) This amounts to saying that 2Cob is generated from five generators, called the unit, counit, multiplication, comultiplication. They include cobordisms: taking the empty set to the circle (the unit)... |

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Citation Context ...cribing a Verity double bicategory is a special case of describing a weak form of higher dimensional categories, or a weak n-category. This broader problem is discussed in more detail by Tom Leinster =-=[Le]-=-, and by Eugenia Cheng and Aaron Lauda [CL]. In light of this more general problem, we can suggest some directions in which to extend this concept further. One is to generalize the concept of a Verity... |

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Citation Context ...se objects are (compact, oriented) (n −1)-dimensional manifolds, and whose morphisms are diffeomorphism classes of compact oriented n-dimensional cobordisms. 2.1. Presentation. It was shown by Abrams =-=[Ab]-=- that 2Cob can be seen as the free symmetric monoidal category on a Frobenius object. (Another good exposition of this was developed by Joachim Kock [Ko].) This amounts to saying that 2Cob is generate... |

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Citation Context ...th corners. Here we shall describe a formalism for describing the ways such cobordisms can be glued together. Louis Crane has written a number of papers on this issue, including one with David Yetter =-=[CY]-=- which describes a bicategory of such cobordisms. Here, we construct a double bicategory, nCob2. One motivation for doing this comes from the fact that interest in nCob has been encouraged by Michael ... |

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Citation Context ...obtained this way nCob2. Having done this, we continue, in section 7 by briefly examining a low-dimensional example, studied in a “decategorified” setting, without using Verity double bicategories in =-=[LP]-=-. This is the case of “open-closed strings”, on which Lauda and Pfeiffer describe a certain kind of TQFT. We see how their category of “open-closed strings” is related to the Verity double bicategory ... |

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Citation Context ...cial case of describing a weak form of higher dimensional categories, or a weak n-category. This broader problem is discussed in more detail by Tom Leinster [Le], and by Eugenia Cheng and Aaron Lauda =-=[CL]-=-. In light of this more general problem, we can suggest some directions in which to extend this concept further. One is to generalize the concept of a Verity double bicategory to a n-tuple bicategory.... |

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Citation Context ...e describe the background for such cobordisms in section 5. Gerd Laures [Lau] discusses the general theory of cobordisms of manifolds with corners. In the terminology used there, introduced by Jänich =-=[Ja]-=-, what we primarily discuss in this paper are 〈2〉-manifolds: in particular, the codimension of the manifold is 2. That is, the manifold M (whose dimension is dim(M) = n) will have a boundary ∂M, which... |

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Citation Context ...ry, just as bicategories were defined by weakening those for a category. The concept of a “weak double category” has been defined (for instance, see Marco Grandis and Robert Paré [GP1]). Thomas Fiore =-=[Fi]-=- describes these as “Pseudo Double Categories”, arising by “categorification” of the theory of categories, and describes examples motivated by conformal field theory. In these cases,�� �� � �� 14 JEF... |

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Citation Context ... where all the bicategories involved are realized as Span(C) for some C. These examples are also analogous to the “profunctor-based examples” of pseudo-double categories described by Grandis and Paré =-=[GP2]-=-. The important example for us here is 2Span(C)0 (we will see the reason for this notation shortly). In Remark 1 we described Bénabou’s demonstration that Obj = Span(C) is a bicategory. There is an an... |

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Citation Context ...s for a double category, just as bicategories were defined by weakening those for a category. The concept of a “weak double category” has been defined (for instance, see Marco Grandis and Robert Paré =-=[GP1]-=-). Thomas Fiore [Fi] describes these as “Pseudo Double Categories”, arising by “categorification” of the theory of categories, and describes examples motivated by conformal field theory. In these case... |

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Citation Context ...ic description of topological quantum field theories, or TQFTs ([At1, At2]). A TQFT assigns a space of states to each manifold, and a linear transformation between states to cobordisms. Ruth Lawrence =-=[Law]-=- described the notion of an extended TQFT. These are theories similar to TQFT’s, for which the theory is defined not on cobordisms, but on manifolds with corners. Crane and Yetter [CY], describe the a... |

2 |
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Citation Context ...gives a bicategory. To finish section 4, we describe a general class of examples of Verity double bicategories, analogous to the result that Span(C) is a bicategory. This was shown by Jean Bénabou in =-=[Be]-=-, who introduced both the concept of a span, and of bicategories. The “double spans” we describe here give a broad class of examples, and in particular, we can use them to derive the fact that there i... |

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1 | Open-Closed Topological Quantum Field Theory and Tangle Homology - Lauda - 2006 |