676,558 research outputs found
Differential Tannakian Categories
We define a differential Tannakian category and show that under a natural
assumption it has a fibre functor. If in addition this category is neutral,
that is, the target category for the fibre functor are finite dimensional
vector spaces over the base field, then it is equivalent to the category of
representations of a (pro-)linear differential algebraic group. Our treatment
of the problem is via differential Hopf algebras and Deligne's fibre functor
construction.Comment: 24 pages; better structured Definition 2 and other statements of the
paper; more examples; more detailed proof of Theorem 1
Cartesian differential categories revisited
We revisit the definition of Cartesian differential categories, showing that
a slightly more general version is useful for a number of reasons. As one
application, we show that these general differential categories are comonadic
over Cartesian categories, so that every Cartesian category has an associated
cofree differential category. We also work out the corresponding results when
the categories involved have restriction structure, and show that these
categories are closed under splitting restriction idempotents.Comment: 17 page
On differential graded categories
Differential graded categories enhance our understanding of triangulated
categories appearing in algebra and geometry. In this survey, we review their
foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen
and Toen-Vaquie.Comment: 30 pages, correction at the end of 3.9, corrections and added
references in 5.
Tangent Categories from the Coalgebras of Differential Categories
Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science
- …