Title: Tangent structure for operadic algebras
In the preface of its Lectures on Noncommutative Geometry [Gin05], Ginzburg writes:
“Each of the mathematical worlds that we study is governed by an appropriate operad. Commutative geometry is governed by the operad of commutative (associative) algebras, while noncommutative geometry ‘in the large’ is governed by the operad of associative not necessarily commutative algebras. […]
There are other geometries arising from operads of Lie algebras, Poisson algebras, etc.“
In this talk we will show that the theory of tangent categories provides a suitable language to make precise this inspiring intuition of Ginzburg: operads produce (algebraic) geometrical theories. We will construct two different tangent categories for each operad, one defined over the category of algebras of the operad and the other one, over the opposite category, using the theories of tangent categories and tangent monads. We will provide concrete examples of these two tangent categories for a few operads. Finally, we will discuss some ideas for future work on this project.
This is joint work with Sacha Ikonicoff and JS Lemay.
- Gin05: Ginzburg; Lectures on Noncommutative Geometry (2005) (https://arxiv.org/pdf/math/0506603.pdf)