The (symmetric) monoidal structure on the category of correspondences

2017 ◽  
pp. 387-417
Keyword(s):  
Monoidal Structure

scholarly journals The monoidal structure on strict polynomial functors

2017 ◽  
Vol 485 ◽  
pp. 213-229 ◽  
Author(s):  
Cosima Aquilino ◽  
Rebecca Reischuk
Keyword(s):  
Monoidal Structure ◽  
Polynomial Functors

scholarly journals Symmetric Monoidal Structure with Local Character is a Property

2019 ◽  
Vol 287 ◽  
pp. 179-190
Author(s):  
Stefano Gogioso ◽  
Dan Marsden ◽  
Bob Coecke
Keyword(s):  
Local Character ◽  
Monoidal Structure

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

scholarly journals A Morphism Double Category and Monoidal Structure

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Saikat Chatterjee ◽  
Amitabha Lahiri ◽  
Ambar N. Sengupta
Keyword(s):  
Monoidal Category ◽  
Vector Spaces ◽  
Monoidal Structure

We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category.

Function spectra

1990 ◽  
Vol 108 (1) ◽  
pp. 31-34 ◽  
Author(s):  
A. D. Elmendorf
Keyword(s):  
Homotopy Theory ◽  
Possible Worlds ◽  
Stable Homotopy ◽  
Stable Category ◽  
Stable Homotopy Theory ◽  
Closed Category ◽  
Monoidal Structure ◽  
Linear Isometries

Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which preserves the symmetric monoidal structure. Since I is not closed, this makes it difficult to see how S might be closed.

scholarly journals The Gray Monoidal Product of Double Categories

2019 ◽  
Vol 28 (3) ◽  
pp. 477-515
Author(s):  
Gabriella Böhm
Keyword(s):  
Practical Significance ◽  
Monoidal Structure ◽  
Double Categories

AbstractThe category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $${\mathbb {A}}$$A, the corresponding internal hom functor "Equation missing" sends a double category $${\mathbb {B}}$$B to the double category whose 0-cells are the double functors $${\mathbb {A}} \rightarrow {\mathbb {B}}$$A→B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

scholarly journals Monoidal structure of the category of u+q-modules

2003 ◽  
Vol 365 ◽  
pp. 183-199 ◽  
Author(s):  
Elísabet Gunnlaugsdóttir
Keyword(s):  
Monoidal Structure

scholarly journals Coherence for monoidal endofunctors

2010 ◽  
Vol 20 (4) ◽  
pp. 523-543 ◽  
Author(s):  
KOSTA DOŠEN ◽  
ZORAN PETRIĆ
Keyword(s):  
Monoidal Category ◽  
Natural Transformation ◽  
Monoidal Categories ◽  
Monoidal Structure ◽  
Natural Transformations ◽  
Finite Products ◽  
Relational Graphs

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, that is, endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the paper, the coherence results are extended to monoidal endofunctors in monoidal categories that have diagonal or codiagonal natural transformations, or where the monoidal structure is given by finite products or coproducts. Monoidal endofunctors are interesting because they stand behind monoidal monads and comonads, for which coherence will be proved in a sequel to this paper.

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