## Homotopical and Higher Algebra

Check in the VVZ for a current information.
 Professor Damien Calaque Assistant Claudia Scheimbauer, Mathieu Anel Time Tue 3-5 Place HG G19.2 Requirements Students are required to give one or two talks, depending on the subject and length. Meetings We will meet once approximately 2 weeks before the talk, usually on Monday afternoons ("office hours") in Damien Calaques office HG G 28.3. Furthermore, you can ask questions to Claudia or Mathieu. First meeting Tue, Sept. 18, 3-5.Talks will be distributed at the first meeting! Please contact the organizers beforehand if you are unavailable on this day.

### Goal

The ultimate goal for this Seminar is to understand the statement of the cobordism hypothesis after Baez-Dolan and Lurie. This will serve as a pretext for introducing several important notions and concepts from higher category theory and homotopy theory.

### Overview of the semester

A detailed outline of the talks can be found here.

##### Schedule
 Week Date Speaker Notes (if available) Week 1 Sept. 25 Merlin The symmetric monoidal category nCob and nTFTs. Week 2 Oct. 2 Daniel, Julian Duality in monoidal categories, Presentation of 1Cob by generators and relations. Week 3 Oct. 9 Thomas 2TFTs and Frobenius algebras. Week 4 Oct. 16 Rade, Damien Extending down TFTs, Bicategories. Week 5 Oct. 23 Claudio, Florian Symmetric monoidal bicategories. Week 6 Oct. 30 summary session Week 7 Nov. 6 Ivan, Camilo 2-dualizable objects. Week 8 Nov. 13 Daniel, Julian Model categories, Models for \infty-groupoids. Week 9 Nov. 20 Adrian Models for (\infty,1)-categories and Quillen equivalences Week 10 Nov. 27 Merlin, Florian The (\infty, 1)-category 1Cob_\infty. Week 11 Dec. 4 Rade, Claudia n-fold Segal spaces, The n-fold complete Segal space Bord_n. Week 12 Dec. 11 Paul Symmetric monoidal structures on higher categories. Week 13 Dec. 18 Mathieu, Damien Around fully dualizable objects in (\infty,n)-categories.

Warning: There are some errors in the talks on Bord_n and its symmetric monoidal structure which will be corrected in an article in preparation by the organizers.

### References

##### Main references
• J. Lurie, On the classification of topological field theories, preprint (2009)
Available here
• J. Baez et J. Dolan, Higher-Dimensional Algebra and Topological Quantum Field Theory, J. Math. Phys. 36 (11), 1995, 6073-6105
Available here
##### Other references
###### Origin of the subject
• M. Atiyah, Topological quantum Field theories, Publ. Math. IHES 68 (1988), 175-186
Available here
###### On the 2-dimensional case
• L. Abrams, Two-dimensional topological quantum Field theories and Frobenius algebras, J. Knot Theory and its Ramifications 5 (1996), 569--587
Available here
• J. Kock, Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts 59, Cambridge University Press, Cambridge, 2004
• C. Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories, PhD Thesis
Available here
###### On bi- and higher categories
• J. Benabou, Introduction to bicategories, In Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47, 1-77, Springer, 1967
• J. Bergner, A survey of $(\infty, 1)$-categories, In Towards Higher Categories, J. Baez and P. May editors, IMA Volumes in Mathematics and its Applications 152. Available here
• J. Bergner, Models for $(\infty,n)$-categories and the cobordism hypothesis, in Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS
Available here
###### On homotopy theory
• P. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics, Birkhauser, Boston, 1999
###### On category theory
• S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer

If you read french, there are some interesting lecture notes from a workshop on Lurie's and Baez-Dolan's work in Paris, which are available here

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