All Questions
Tagged with monoidal-categories abelian-categories
34 questions
1
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38
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Is a direct summand of a dualisable object itself dualisable?
If an object $X$ in an abelian monoidal category is the direct summand of a dualisable object, is $X$ itself dualisable? This is true in the category of modules over a commutative ring, since then a ...
- 2,133
1
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0
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52
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Associtivity of Tensor Product of Modules Over Algebras in a Tensor Category
I am attempting to prove that modules over a commutative algebra (monoid) $A$ in a fixed tensor category $\mathcal{T}$ form a tensor category $\mathcal{T}_A$. All of the references I have found say it ...
11
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3
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1k
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How much data does a category contain?
This might seem like a very vague question, but the details are really confusing me. So, for example, say we are studying the category of $A$-modules $\mathsf{Mod}_A$ where $A$ is a commutative unital ...
- 407
6
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1
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108
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Is there a notion of tensor-completion of a category?
Given an additive category $\mathcal{C}$, sometimes one wants to consider its Karoubi envelope or idempotent completion $\mathrm{Kar}(\mathcal{C})$, whose objects are summands of objects in $\mathcal{...
- 1,923
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37
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Does the currying of a closed abelian monoidal category preserves addition?
Let $V$ be a left-closed abelian monoidal category. That is, $V$ is a left-closed monoidal category which is also an abelian category. Let $\Phi:\hom(y\otimes x,z)\to\hom(x,[y,z])$ be the currying of $...
- 1,573
3
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1
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427
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Deligne’s tensor product of algebra module categories
$A$ and $B$ are finite dimensional $\mathbb{k}$-algebras. $\textrm{Mod}_A$ is the category of finite dimensional $A$-modules.
In Proposition 1.46.2. of the note，it is claimed that $\textrm{Mod}_{A\...
- 83
3
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1
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202
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Isomorphism between $\operatorname{End}(F\otimes F)$ and $\operatorname{End}(F)\otimes \operatorname{End}(F)$, where F is an exact faithful functor.
Let $\mathcal{C}$ be a finite $k$-linear abelian category, and $\operatorname{Vec}$ be the category of finite dimensional vector spaces over $k$. Let $F_1,\ F_2:\ \mathcal{C}\rightarrow \operatorname{...
3
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1
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97
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About "6j symbol": How to understand a vertor space tensor with an object in a tensor category?
Let $\mathcal{C}$ be a semisimple (multi)tensor categroy over field $k$, with simple objects $\{V_i\}_{i\in I}$. We define
$$
H_{i, j}^{\ell}=\operatorname{Hom}_\mathcal{C}\left(V_{\ell}, V_{i} \...
0
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0
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95
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Direct sum and short exact sequence, as well as, tensor product and what?
In an abelian category, the notion of direct sum is generalized by the notion of short exact sequence (see split exact sequence).
Question: In a monoidal category, can the notion of tensor product be ...
- 5,991
2
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0
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107
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Objects in Ind-category are filtered colimits of compact subobjects
In his paper 'Categories Tensorielles' in section 2.2 Deligne states that if in a tensor category $\mathcal{A}$ all objects are of finite length, then every object of the Ind-category $\text{Ind}\...
- 1,190
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126
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Why do we need $End(I)=k$ for neutral Tannakian categories?
I have been reading Milne's book 'Basic Theory of Affine Group Schemes', and in particular the section on Tannaka duality for affine group schemes. The 'final' theorem of this section is displayed ...
- 1,190
0
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2
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374
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Morphisms of a category can be regarded as the objects of a group?
Here is a category with a collection of objects $$A, B, C$$ and collection of morphisms denoted $$f, \quad g, \quad g ∘ f,$$ and the loops are the identity arrows. This category is typically denoted ...
- 3,787
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1
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94
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The implication that the small category does not need to have TOTALITY (closure)?
What is the implication and the importance of demanding that the small category does not need to have TOTALITY?
Although it is not obvious to me, but it seems that TOTALITY iff closure (can you also ...
- 3,787
7
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1
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252
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A lemma in Tensor Categories (Etingof et al)
Lemma 8.10.5 in EGNO's Tensor Categories basically states
Let $\mathcal{C}$ be a tensor category over an algebraically closed field $\mathbb{k}$ with braiding $c$.
For any nonzero simple object $X$ ...
- 2,115
1
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1
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109
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Projective cover of simple counts occurences of simple in composition series?
On page 11 in Tensor Categories is the statement
Let $\mathcal{C}$ be a finite abelian $k$-linear category. Then for any $X,Y \in \mathcal{C}$ with $X$ simple we have
\begin{align*}
\dim_k \...
- 21
1
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0
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65
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Additivization of functors in an abelian monoidal category
Crossposted on MathOverflow here.
I'm having trouble with the proof of Lemma 2.9 in "Cohomology of Monoids in Monoidal Categories" by Baues, Jibladze, and Tonks, and I was wondering if someone could ...
- 1,446
5
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1
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148
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How to show that the category of finite vector spaces is indecomposable?
In [1] it is said that the category $\mathbf{FinVect}$ of finite vector spaces is a tensor category. I am trying to convince myself that this is indeed the case.
One of the properties that $\mathbf{...
- 2,794
4
votes
1
answer
79
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A monoidal category that preserves subobjects
Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
- 1,570
5
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1
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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
0
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1
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241
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In a tensor category, does $X\otimes Y\cong 0$ imply $Y\cong 0$ for non-zero $X$?
By a tensor category I mean a locally finite rigid $k$-linear abelian category with bilinear tensor product, and such that $\operatorname{Hom}(1,1)\cong k$.$^1$
Suppose we fix some non-zero object $...
- 2,115
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1
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67
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Smallest abelian braided monoidal subcategory containing an object $V$
Let $\mathcal{C}$ be an abelian braided monoidal category with countable direct sums compatible with the tensor product (i.e. $X\otimes \bigoplus_{i \in \mathbb{N}} V_i \cong \bigoplus_{i \in \mathbb{...
- 1,570
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0
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58
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the Verlinde formula
The Verlinde formula writes the fusion coefficient in terms of S matrix. My question is that for fusion category without braiding, is there a similar formula which gives the fusion coefficient in ...
- 21
3
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1
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130
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Are there "distributive bicategories"?
In the bicategory $\mathsf{Bimod}$ of rings, bimodules and bimodule morphisms there is also a direct sum making every category $\mathsf{Bimod}(\mathsf{R},\mathsf{S})$ of $(\mathsf{R},\mathsf{S})$-...
- 163
2
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1
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290
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Difference between monoidal and tensor categories
Is a monoidal category just another word for tensor category or are those two different (but still similiar) things in the sense that one of them is more general?
Are those categories supposed to be ...
user372565
9
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0
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932
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What's there to do in category theory?
I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it.
Now ...
- 44.4k
6
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148
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Reconstruction of monoidal categories
Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator).
Bruce Westbury ...
- 12.8k
5
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1
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141
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Completeness of 2-category of Monoidal Categories
Is the 2-category of monoidal categories complete?
If not, can any conditions be imposed to satisfy completeness?
- 685
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76
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Does Projectiveness always imply flatness?
I know that a project module is always flat, deduced form the properties and abundance of free modules. I'm trying to figure out how essential role the free modules play in this result. So I'd like to ...
- 6,015
3
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1
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182
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Can tensor abelian categories always be embedded into the category of modules?
Let $(\mathcal A, +,\otimes,I)$ a small symmetric monoidal abelian category. I know that $\mathcal A$ can be embedded into the category of $R$-module for a certain ring $R$. But can we make such ...
- 6,015
5
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1
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109
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When modular tensor categories are equivalent?
I would like to know when we say that two modular tensor categories are equivalent.
Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
user65175
6
votes
1
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562
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Tensor products and morphisms
Let $C$ be semisimple category with simple objects $X_1, \dots, X_r$.
Suppose we have a fusion relation $X_i\otimes X_j =\bigoplus_{l=1}^r N_{ij}^l X_l$.
Let $m\in \mathbb{N}$ and let $g:mX_j \to ...
user65175
1
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0
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56
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Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring
Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2).
Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
- 519
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1
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84
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Does "modular category" make sense without saying "abelian" or "linear"?
I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a ...
- 2,692
3
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1
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233
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The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?
The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$
Intuitively, I would have defined the ...
user153312