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Archive for the ‘homological algebra’ Category

The goal of today’s post is to introduce and discuss semiadditive categories. Roughly speaking, these are categories in which one can add both objects and morphisms. Prominent examples include the abelian categories appearing in homological algebra, such as categories of sheaves and modules and categories of chain complexes.

Semiadditive categories display some interesting categorical features, such as the prominence of pairs of universal properties and the surprising ways in which commutative monoid structures arise, which seem to be underemphasized in textbook treatments and which I would like to emphasize here. I would also like to emphasize that their most important properties are unrelated to the ability to subtract morphisms which is provided in an additive category.

In this post, for convenience all categories will be locally small (that is, \text{Set}-enriched).

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In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra A (the algebra of observables of some quantum system) and a Hamiltonian H \in A, we obtain a derivation [-, H], which is (up to some scalar multiple) the infinitesimal generator of time evolution. This is a natural and general way to start with an algebra and an energy function and get a notion of time evolution which automatically satisfies conservation of energy.

However, if A is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian H \in A and produces a notion of time evolution. How does that work? It turns out that for algebras of observables A of a classical system, we can think of A as the classical limit \hbar \to 0 of a family A_{\hbar} of noncommutative algebras. While A is commutative, the noncommutativity of the family A_{\hbar} equips A with the extra structure of a Poisson bracket, and it is this Poisson bracket which allows us to describe time evolution.

Today we’ll describe one way to formalize the notion of taking the classical limit using the deformation theory of algebras. We’ll see how Poisson brackets pop out along the way, as well as the relevance of the lower Hochschild cohomology groups.

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