# Simple transitive 2-representations via (co)algebra 1-morphisms

```@article{Mackaay2016SimpleT2,
title={Simple transitive 2-representations via (co)algebra 1-morphisms},
author={Marco Mackaay and Volodymyr Mazorchuk and Vanessa Miemietz and Daniel Tubbenhauer},
journal={Indiana University Mathematics Journal},
year={2016}
}```
• Published 19 December 2016
• Mathematics
• Indiana University Mathematics Journal
For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples explicitly.

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## References

SHOWING 1-10 OF 55 REFERENCES

### Transitive 2-representations of finitary 2-categories

• Mathematics
• 2014
In this article, we define and study the class of simple transitive \$2\$-representations of finitary \$2\$-categories. We prove a weak version of the classical Jordan-H{\"o}lder Theorem where the weak

### Two-color Soergel Calculus and Simple Transitive 2-representations

• Mathematics
• 2019
Abstract In this paper, we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In

### Additive versus abelian 2-representations of fiat 2-categories

• Mathematics
• 2011
We study connections between additive and abelian 2-rep- resentations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell

### Isotypic faithful 2-representations of J-simple fiat 2-categories

• Mathematics
• 2014
We introduce the class of isotypic 2-representations for ﬁnitary 2-categories and the notion of inﬂation of 2-representations. Under some natural assumptions we show that isotypic 2-representations

### Module categories, weak Hopf algebras and modular invariants

AbstractWe develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal

### Cell 2-representations of finitary 2-categories

• Mathematics
Compositio Mathematica
• 2011
Abstract We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and

### Simple Transitive 2-Representations for Two Nonfiat 2-Categories of Projective Functors

• Mathematics
Ukrainian Mathematical Journal
• 2019
We show that any simple transitive \$2\$-representation of the \$2\$-ca\-te\-go\-ry of projective endofunctors for the quiver algebra of \$\Bbbk(\xymatrix{\bullet\ar[r]&\bullet})\$ and for the quiver

### Representations of tensor categories and Dynkin diagrams

• Mathematics
• 1994
In this note we illustrate by a few examples the general principle: interesting algebras and representations defined over Z_+ come from category theory, and are best understood when their categorical