A model-categorical cotangent complex formalism (part 1)
Event details
| Date | 22.03.2016 |
| Hour | 10:15 › 11:30 |
| Speaker |
Matan Prasma (Radboud Universiteit Nijmegen) |
| Location |
CM113
|
| Category | Conferences - Seminars |
One of the first applications of the theory of model categories was Quillen homology. Building on the notion of Beck modules, one defines the cotangent complex of an associative or commutative (dg)-algebra as the derived functor of its abelianization. The latter is a (bi)module over the original algebra, and its homology groups are called the (Andre'-)Quillen homology. The caveat of this approach is that the cotangent complex is not defined as a functor on the category of all algebras but rather on a fixed slice thereof. To remedy this, Lurie's "cotangent complex formalism" (Higher Algebra & 7) uses the infinity-categorical Grothendieck construction and gives a general "global" treatment for the cotangent complex of an algebra over a (coherent) infinity-operad.
In this talk I will propose a way to parallel Lurie's formalism using model categories which is based on the model-categorical Grothendieck construction as developed by Yonatan Harpaz and myself. We will see how to define the tangent model category of a model category, accompanied with an "abstract" cotangent complex functor. We will then identify the tangent category of algebras over a dg-operad, as the category of operadic algebras and their modules and the abstract cotangent complex with the operadic cotangent complex. This latter result is an extension of the results in Higher Algebra (and following Bastera-Mandell) to the dg-case. At the cost of potentially restricting generality, our approach offers a simplification to that of Lurie's in that one can avoid carrying a significant amount of coherent data.
I will assume basic familiarity with (model) categories but not much more.
This is a joint work with Yonatan Harpaz and Joost Nuiten.
In this talk I will propose a way to parallel Lurie's formalism using model categories which is based on the model-categorical Grothendieck construction as developed by Yonatan Harpaz and myself. We will see how to define the tangent model category of a model category, accompanied with an "abstract" cotangent complex functor. We will then identify the tangent category of algebras over a dg-operad, as the category of operadic algebras and their modules and the abstract cotangent complex with the operadic cotangent complex. This latter result is an extension of the results in Higher Algebra (and following Bastera-Mandell) to the dg-case. At the cost of potentially restricting generality, our approach offers a simplification to that of Lurie's in that one can avoid carrying a significant amount of coherent data.
I will assume basic familiarity with (model) categories but not much more.
This is a joint work with Yonatan Harpaz and Joost Nuiten.
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Practical information
- Informed public
- Free
Organizer
- Magdalena Kedziorek