# Tagged Questions

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### Pivotal functors of that are substantially different from finite group homomorphisms

Fusion categories can be seen as generalisations of the representation category of finite groups. I'm interested in spherical fusion categories. I'm trying to find "interesting" functors from a ...
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### Serre functor of a subcategory (in particular parabolic category O)

For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms $$Hom(A, S(B)) \cong Hom(B, A)^*$$ ...
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### When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...
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### Are there natural examples of non-symmetric Frobenius algebras?

Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in ...
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### An exact sequence which does not split

Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism. Suppose that it is ...
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### From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...
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### Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
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### History and motivation for Tannaka, Krein, Grothendieck, Deligne et al. works on Tannaka-Krein theory?

I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask: Question: What was the motivation and historical context for ...
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### Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...
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### Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...
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### Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
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### What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of $\mathfrak{g}$-mod?

This question may be trivial for experts. Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, ...
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### U(1) vs. BZ and representations of 2-groups

$U(1)$ seems to lead a dual life. On one hand it is the group we know and love, and on the other, it is the classifying space of the integers. Thinking about $n$-groups says that we should also think ...
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### An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
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### What's an example of a locally presentable category “in nature” that's not $\aleph_0$-locally presentable?

Recall the notion of locally presentable category (nLab): $\DeclareMathOperator{\Hom}{Hom}$ Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less ...
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### What is known about higher-categorical reconstruction theorems? (reference request)

The answer to my question is almost certainly "not much" — at least, I've asked a few people, and that was their answer. But I'd like to refine this answer, and MathOverflow seems like the best ...
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### What makes the stable module category stable?

When geometrically flavoured words like "mapping cone" or "chain homotopy" crop up in homological algebra, there's usually a good reason. (In this case, looking at the chain complexes associated with ...
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### Looking for references talking about category of topological vector spaces

It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...
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### When are modules and representations not the same thing?

I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind: A ring ...