# Pretriangulated 2-representations via dg algebra 1-morphisms

@inproceedings{Laugwitz2022Pretriangulated2V, title={Pretriangulated 2-representations via dg algebra 1-morphisms}, author={Robert Laugwitz and Vanessa Miemietz}, year={2022} }

. This paper develops a theory of pretriangulated 2 -representations of dg 2 -categories. We characterize cyclic pretriangulated 2 -representations, un- der certain compactness assumptions, in terms of modules over dg algebra 1 morphisms internal to associated dg 2 -categories of compact dg modules. Fur- ther, we investigate the Morita theory and quasi-equivalences for such dg 2 representations. We relate this theory to various classes of examples of dg cate- goriﬁcations from the literature.

## One Citation

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## References

SHOWING 1-10 OF 56 REFERENCES

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

- MathematicsIndiana University Mathematics Journal
- 2019

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…

Cell 2-representations of finitary 2-categories

- MathematicsCompositio 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…

Grothendieck ring of pretriangulated categories

- Mathematics
- 2004

We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal decompositions of corresponding triangulated…

Picard Groups for Derived Module Categories

- Mathematics
- 2003

In this paper we introduce a generalization of Picard groups to derived categories of algebras. First we study general properties of this group. Then we consider easy particular algebras such as…

Deriving DG categories

- Mathematics
- 1994

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5],…

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…

Heisenberg categorification and Hilbert schemes

- Mathematics
- 2012

Given a finite subgroup G of SL(2,C) we define an additive 2-category H^G whose Grothendieck group is isomorphic to an integral form of the Heisenberg algebra. We construct an action of H^G on…

Categorification of tensor powers of the vector representation of U q (gl(1j1))

- Mathematics
- 2014

We consider the monoidal subcategory of finite-dimensional representations of Uq(gl(1|1)) generated by the vector representation, and we provide a graphical calculus for the intertwining operators,…

Spherical DG-functors

- Mathematics
- 2013

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B. We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A). We give…

DG structures on odd categorified quantum $sl(2)$

- Mathematics
- 2018

We equip Ellis and Brundan's version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading…