Skew-monoidal categories and bialgebroids

By Korn'el Szlach'anyi. 2012

92 citations recorded by the source corpus.

Abstract Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R -modules are precisely the right bialgebroids over the ring R . These skew-monoidal structures induce quotient skew-monoidal structures on the category of R – R -bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law. Pursuing a Theorem of Day and Street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46c78aa8914d5c0de75d926dd380f622fbeea365

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Ross Street
2. Michael Batanin
3. Stephen Lack
4. Bob Coecke
5. C. Berger

In Geometry and Topology

1. Ross Street
2. Stephen Lack
3. Michael Batanin
4. Mark Weber
5. Aaron D. Lauda
6. Bob Coecke
7. C. Berger
8. Joachim Kock

In Mathematical Physics

1. Ross Street
2. Stephen Lack
3. Michael Batanin
4. Mark Weber
5. Aaron D. Lauda
6. Bob Coecke
7. C. Berger
8. Joachim Kock

Citations in the corpus

• Skew monoidales, skew warpings and quantum categories (2012, unknown): “fficed. Kelly [16] reduced these to two axioms. However, the reduction depends critically on invertibility of the associativity and unit constraints. A skew-monoidal category in the sense of Szlachányi [28] is defined in the same way as a monoidal category (see [11] or [22]) except that the associativi”
• Skew-closed categories (2012, background): “ivity and unit constraints. The interest back then was quite pedagogical: the invertibility of the associativity constraint of the monoidal structure was at odds with the closed structure. In view of [14] and [9], the topic seems to have a renewed interest. Most of the ideas and calculations here ar”
• Monads need not be endofunctors (2010, background): “[33] K. Szlachányi.”