271 research outputs found
Stacks of Ann-Categories and their morphisms
We show that -categories admit a presentation by crossed
bimodules, and prove that morphisms between them can be expressed by special
kinds spans between the presentations. More precisely, we prove the groupoid of
morphisms between two -categories is equivalent to that of
bimodule butterflies between the presentations. A bimodule butterfly is a
specialization of a butterfly, i.e. a special kind of span or fraction, between
the underlying complexesComment: 23 pages. One added section on the class of a stack of ann-categories
and Shukla cohomology. One appendix includes an argument courtesy of T.
Pirashvili showing the equivalence between Shukla and Andr\'e-Quillen
cohomology for associative algebras, when Shukla cohomology is defined via a
model structure on DGA
Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings
We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in
each variable, a nonabelian version of a biextension. We show that such a
biextension satisfies additional triviality conditions which make it a bilinear
analog of the kind of spans known as butterflies and, conversely, these data
determine a bimonoidal functor. We extend this result to -variables, and
prove that, in a manner analogous to that of butterflies, these
multi-extensions can be composed. This is phrased in terms of a multilinear
functor calculus in a bicategory. As an application, we study a bimonoidal
category or stack, treating the multiplicative structure as a bimonoidal
functor with respect to the additive one. In the context of the multilinear
functor calculus, we view the bimonoidal structure as an instance of the
general notion of pseudo-monoid. We show that when the structure is ring-like,
i.e. the pseudo-monoid is a stack whose fibers are categorical rings, we can
recover the classification by the third Mac Lane cohomology of a ring with
values in a bimodule.Comment: Accepted version to appear in Theory and Applications of Categories;
61 Pages; the new Appendix E contains the full hypercohomology computation of
the characteristic class of a ring-like stac
Rational motivic path spaces and Kim's relative unipotent section conjecture
We initiate a study of path spaces in the nascent context of "motivic dga's",
under development in doctoral work by Gabriella Guzman. This enables us to
reconstruct the unipotent fundamental group of a pointed scheme from the
associated augmented motivic dga, and provides us with a factorization of Kim's
relative unipotent section conjecture into several smaller conjectures with a
homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that
the path spaces of the punctured projective line over a number field are
concentrated in degree zero with respect to Levine's t-structure for mixed Tate
motives. This constitutes a step in the direction of Kim's conjecture.Comment: Minor corrections, details added, and major improvements to
exposition throughout. 52 page
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