All Questions
Tagged with monoidal-categories duality-theorems
19 questions
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If $A$ and $B$ are dualizable objects in a monoidal category, is the unit of the one duality the inverse of the counit of the other duality?
I'm currently trying to wrap my head around dualizable objects in monoidal categories and I was wondering whether the following claim holds:
Let $A$ and $B$ be dualizable objects in a monoidal ...
- 411
2
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0
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123
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Do symmetric monoidal functors preserve reflexivity?
Suppose that $(C, \otimes_C, \mathcal{H}om_C, I_C), (D, \otimes_D, \mathcal{H}om_D, I_D)$ are closed symmetric monoidal categories. Let
$$
F: (C, \otimes_C, I_C) \rightarrow (D, \otimes_D, I_D)
$$
...
- 1,604
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84
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Dual objects where only one zig-zag identity holds?
Recall the definition of a (left) dual object in a monoidal category.
If one requires that both the evaluation and the coevaluation are isomorphisms, one zig-zag-identity implies the other (see here).
...
- 3,158
5
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Examples of self-dual categories
Call a category $C$ self-dual if there exists an equivalence of categories $F: C \rightarrow C^{op}$. I am looking for examples of self-dual categories. Rigid monoidal categories and more generally ...
- 3,158
2
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64
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Cartesian monoidal star-autonomous categories
EDIT: Note that I have cross-posted the question on MathOverflow in the meantime.
1. Question
Any rigid cartesian monoidal category is trivial (see here). Star-autonomity is a generalization of ...
- 3,158
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185
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Adjunctions and dualizable objects in symmetric monoidal category
Let $(C, \otimes, I)$ be a symmetric monoidal category (for example, modules over a commutative ring), and let $M$ be a dualizable object with dual $M^\vee$.
Then $\_ \otimes M$ is left adjoint to $\...
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49
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Is existence of adjunction between $a\otimes{-}$ and $b\otimes{-}$ enough for duality?
Let $(\mathfrak{C},\otimes,1)$ be a monoidal category. An object $a$ is dual to $b$ if there exist evaluation $ev\colon a\otimes b\to 1$ and coevaluation $coev\colon 1\to b\otimes a$ morphisms ...
- 125
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Star-autonomous categories are linearly distributive categories with negation?
1.Context
On page 28 of Weakly distributive categories Cockett and Seely are trying to prove the following statement:
The notions of symmetric weakly distributive categories with negation and star-...
- 3,158
6
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278
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Graphical calculus for star-autonomous categories?
1. Definiton
Let $(C, \otimes, I, a, l,r)$ be a (not necessarily symmetric) monoidal category.
A (planar) star-autonomous structure on the monoidal category $C$ consists of an adjoint equivalence $D \...
- 3,158
3
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Dual of braiding in symmetric monoidal category
Let $(\mathcal{C},\otimes,1)$ be a symmetric monoidal category with symmetry $\gamma$.
Assume $X,Y\in\mathcal{C}$ are a dual pair in the sense that there exist evaluation and coevaluation morphisms $...
- 125
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77
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Left versus right dualizability in a braided monoidal category
Let's say we have a duality between two objects $X$ and $Y$ in a braided monoidal category given by $c \colon \mathbb 1 \to X \otimes Y$, $e \colon Y \otimes X \to \mathbb 1$. I was wondering whether ...
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144
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Self-dual object in tensor triangulated categories which are not strongly dualizable
I am working in a closed symmetric tensor triangulated category. This is a triangulated category $\mathcal{T}$ admitting a symmetric monoidal structure with tensor product $\otimes$ which is closed. ...
- 2,119
1
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138
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What are the dualizable objects in the category of Hilbert spaces?
Let $\mathbf{Hilb}$ be the category of Hilbert spaces and continuous linear maps. Turn it into a symmetric monoidal category using the tensor product of Hilbert spaces. What are the dualizable objects?...
- 1,298
3
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Going from the unit-counit definition of the dual pair to the Hom-adjunction
In "Categories and Sheaves" by Kashiwara and Schapira on the page 101, the definition of the dual pair in tensor category is given and in the next theorem it is shown that if $(X,Y)$ is a ...
- 549
4
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Tannaka duality for closed monoidal categories
The nLab article on the Tannaka duality says that this theory was generalized to monoids in arbitrary closed monoidal category (symmetric and complete in some sense): if we take a monoid $A$ in such a ...
- 2,483
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Dualizable presheaves with respect to Day convolution
Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, Set)$ its category of presheaves regarded as a closed symmetric monoidal category via Day ...
- 4,461
5
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309
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Is the class of dualizable objects in an abelian monoidal category closed under sums, kernels and cokernels?
Goodmorning to everybody.
I am in the following situation.
I have been told that in an abelian monoidal category (I assume this means an abelian category $\mathscr{A}$ with a monoidal structure $(\...
- 2,942
1
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0
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182
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Examples of weakly dualizable objects in a non-closed monoidal category.
The following is a straightforward generalization of the notion of dualizable object in a symmetric monoidal category given in Duality, Trace and Transfer by Albrecht Dold and Dieter Puppe to non-...
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2
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Connections of Finite groups and quantum groups
I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
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