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Register a new userThe Algebroid of a Groupoid in a Tangent Category
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Authors
Matthew Burke
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We generalise the construction of the Lie algebroid of a Lie groupoid so that it can be carried out in any tangent category. First we reconstruct the bijection between left invariant vector fields and source constant tangent vectors based at an identity element for a groupoid in a category equipped with an endofunctor that has a retraction onto the identity functor. Second we use the full structure of a tangent category to construct the algebroid of a groupoid. Finally we show how the classical result concerning the splitting of the tangent bundle of a Lie group can be carried out for any pregroupoid.
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By a ring groupoid we mean an animated ring whose i-th homotopy groups are zero for all i>1. In this expository note we give an elementary treatment of the (2,1)-category of ring groupoids (i.e., without referring to general animated rings and without using n-categories for n>2). The note is motivated by the fact that ring stacks play a central role in the Bhatt-Lurie approach to prismatic cohomology.
We make some beginning observations about the category $mathbb{E}mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that of $S$-equivalence classes, which is induced by a computable function. We also consider some full subcategories of $mathbb{E}mathrm{q}$, such as the category $mathbb{E}mathrm{q}(Sigma^0_1)$ of computably enumerable equivalence relations (called ceers), the category $mathbb{E}mathrm{q}(Pi^0_1)$ of co-computably enumerable equivalence relations, and the category $mathbb{E}mathrm{q}(mathrm{Dark}^*)$ whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in $mathbb{E}mathrm{q}(Sigma^0_1)$ the epimorphisms coincide with the onto morphisms, but in $mathbb{E}mathrm{q}(Pi^0_1)$ there are epimorphisms that are not onto. Moreover, $mathbb{E}mathrm{q}$, $mathbb{E}mathrm{q}(Sigma^0_1)$, and $mathbb{E}mathrm{q}(mathrm{Dark}^*)$ are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in $mathbb{E}mathrm{q}(Pi^0_1)$ whose coequalizer in $mathbb{E}mathrm{q}$ is not an object of $mathbb{E}mathrm{q}(Pi^0_1)$.
We show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. We prove that Penon tricategories cannot give all tricategories, but we show that a slightly modified version of the definition rectifies the situation. We give the modified definition, using non-reflexive rather than reflexive globular sets, and show that the problem with doubly degenerate tricategories does not arise.
We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an involution algebroid and every involution algebroid in a tangent category admits a Lie bracket on the sections of its underlying bundle. As an illustrative application we take the first steps in developing the homotopy theory of involution algebroids.
This preprint contains a part of the results of our earlier preprint arXiv:0907.3335v2 presented in a form suitable for journal publication. It covers a construction of a 2-fold monoidal structure on the category of tetramodules, with all necessary definitions, and an overview of the results of R.Taillefer [Tai1,2] on tetramodules and the Gerstenhaber-Schack cohomology [GS] (formerly served as Appendix in arXiv:0907.3335v2), as well as a computation of the Gerstenhaber-Schack cohomology for the free commutative cocommutative bialgebra S(V), for a V is a vector space.
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