## On the cobordism and commutative monoid with cancellation approaches to . . . (2007)

### BibTeX

@MISC{Fiore07onthe,

author = {Thomas M. Fiore},

title = {On the cobordism and commutative monoid with cancellation approaches to . . . },

year = {2007}

}

### OpenURL

### Abstract

### Citations

242 |
Braided tensor categories
- Joyal, Street
- 1983
(Show Context)
Citation Context ...nius condition. I thank John Baez for pointing out that the Frobenius condition in a strict symmetric monoidal category is equivalent to the requirement that each object is its own dual as defined in =-=[7]-=-. Self-duality can be replaced by a weaker condition that requires the existence of duals, but then the axioms would have to be modified in subtle ways. The present definition with self-duality is suf... |

73 |
The Definition Of Conformal Field Theory
- Segal
- 2004
(Show Context)
Citation Context ...e 2-category of pseudo 2-algebras over a fixed theory in certain situations. c○ 2006 Elsevier B.V. All rights reserved. MSC: Primary: 81T40; 18D05; secondary: 18C10 1. Introduction In the foreword to =-=[11]-=-, Graeme Segal points out that conformal field theory with one dimensional anomaly can be axiomatized using the ‘cobordism category’ where objects are 1-dimensional manifolds and morphisms are worldsh... |

71 |
Categories for the Working
- Lane
- 1998
(Show Context)
Citation Context ...his paper. Appendix A. Bicategories, theories, 2-theories, and pseudo algebras The basics of bicategories can be found in the original paper [1], in the concise account [9], or in the last chapter of =-=[10]-=-. A thorough description of theories, 2-theories, and pseudo algebras can be found in [3]. Theories were introduced in [8], while 2-theories and pseudo algebras over theories and 2-theories were intro... |

37 | Conformal field theory and elliptic cohomology
- Hu, Kriz
(Show Context)
Citation Context ...s all boundary components on equal footing, and uses trace instead of compositions to axiomatize gluing. This symmetric approach was in fact only outlined in [11], and was made completely rigorous in =-=[3,5,6]-=-. Unlike the cobordism approach, the older approach axiomatizes conformal field theory with n-dimensional modular functor. In this paper, I shall point out certain more subtle points of the relationsh... |

29 |
Lawvere, Functorial Semantics of Algebraic Theories
- William
- 1963
(Show Context)
Citation Context ...lly equally technical. The completely rigorous approach to conformal field theory of [3,5,6] relies on various notions of category theory. The first ingredients are the notion of Lawvere theory as in =-=[8]-=- and its generalization called 2-theory in [5]. Theories are used to describe algebraic structures on a set, such as a commutative monoid. On the other hand, 2-theories are ∗ Tel.: +1 773 702 0088; fa... |

26 | Closed and open conformal field theories and their anomalies
- Hu, Kriz
(Show Context)
Citation Context ...aper then states that the resulting structure of ‘stack of pseudo Frobenius symmetric monoidal categories’ implies the structure of ‘stack of pseudo commutative monoids with cancellation’ (SPCMC2) of =-=[5,6]-=-, which is the basic structure of the symmetric approach of [3,5,6]. Since we are speaking of chiral conformal field theory, the Grothendieck site for these stacks consists of the category of finite d... |

25 |
Structures algébriques dans les catégories, Cahiers Topologie Géom. Différentielle 10
- Benabou
- 1968
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Citation Context ... has two inbound boundary components with labels (b, 1) and (b ′ , 1) respectively, then those two boundary components are glued according to f (b,1) (z) ∼ f (b′ ,1) (φ(z)) for all z ∈ S1 , where φ : =-=[0, 1]-=- �� ∐ S1 is the map t ↦→ exp(−2πit) . We then remove α from y x (we conceive of α as being glued between the two boundary components).� 598 T.M. Fiore / Journal of Pure and Applied Algebra 209 (2007)... |

18 | Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
- Fiore
(Show Context)
Citation Context ...s all boundary components on equal footing, and uses trace instead of compositions to axiomatize gluing. This symmetric approach was in fact only outlined in [11], and was made completely rigorous in =-=[3,5,6]-=-. Unlike the cobordism approach, the older approach axiomatizes conformal field theory with n-dimensional modular functor. In this paper, I shall point out certain more subtle points of the relationsh... |

13 | Basic bicategories
- Leinster
(Show Context)
Citation Context ...ents on an earlier version of this paper. Appendix A. Bicategories, theories, 2-theories, and pseudo algebras The basics of bicategories can be found in the original paper [1], in the concise account =-=[9]-=-, or in the last chapter of [10]. A thorough description of theories, 2-theories, and pseudo algebras can be found in [3]. Theories were introduced in [8], while 2-theories and pseudo algebras over th... |

1 |
Pseudo algebras and double categories
- Fiore
- 2006
(Show Context)
Citation Context ...he algebraic structure in the cobordism approach. A strict Frobenius symmetric monoidal category gives rise to a strict commutative monoid with cancellation (CMC) via a map of 2-theories. As noted in =-=[2]-=-, a pseudo category is not a bicategory nor a double category. Therefore in passing from strict Frobenius symmetric monoidal categories to 1 In [5,6] the term ‘lax’ is used to mean up to coherence iso... |

1 |
The Jacobian of a rigged surface (in preparation
- Fiore, Kriz
(Show Context)
Citation Context ..., H) of stacks of pseudo commutative monoids with cancellation. One can also take X to be other pseudo commutative monoids, for example, the pseudo commutative monoid of Jacobians with boundary as in =-=[4]-=-. 6.2. The cobordism approach to conformal field theory As we have seen in Theorems 5.3 and 5.4, a pseudo algebra over the cobordism 2-theory gives rise to a pseudo commutative monoid with cancellatio... |