# Questions tagged [morita-equivalence]

Two rings are said to be Morita equivalent if their categories of (left) modules are equivalent. The notion is also used in more general contexts when certain categories of representations are equivalent.

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### Morita equivalence for monoidal categories

I came across this MO post and it got me thinking. In monoidal categories we can define module and bi-module categories, so what can we say about morita equivalence in this situation? Which of the ...
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### Can a braided fusion category have an order-2 Morita equivalence class which cannot be simultaneously connected and isomorphic to its opposite?

Let $\mathcal{B}$ be a braided fusion category over $\mathbb{C}$. Let me write $\mathrm{Alg}(\mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $\mathcal{B}$, ...
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### Categorical Morita equivalence implies equivalence of module categories?

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true ($R$-Mod) $\simeq$ ($S$-Mod). $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective ...
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### Picard-surjectivity and Morita-equivalence

Let us say that an algebra $A$ over a field $k$ is Picard-surjective if the canonical map $$\mathrm{Aut}(A) \rightarrow \mathrm{Pic}(A)$$ is surjective. Here $\mathrm{Pic}(A)$ denotes the group of ...
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### Does Morita theory hint higher modules for noncommutative ring?

Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. ...
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### Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...
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### Algebras Morita equivalent to their centers

Hi, I wonder if there is a name for: 1) Algebras which are Morita equivalent to their centers, or 2) dg-algebras which are derived Morita equivalent to their Hochshild cohomology? For instance, (...
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### Morita equivalence for compact groups

Let $K$ be a compact or a finite group with a closed subgroup $H$. Let $C(K)$ be the convolution algebra of continuous functions on $K$. The Peter Weyl theorem asserts that the $*$ algebra $C(K)$ and ...
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### Morita equivalence of DG algebras? (reference needed)

A stupid question: whether ther exists a universally recognized definition of when two differential graded algebras should be Morita equivalent? I mean a sort of equivalence which would incorporate ...
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### Compare three 2-categories of (Lie) groupoids

Lie groupoids are groupoids with smooth structures. There is a nature 2-category of Lie groupoids: Lie groupoids, smooth functors of Lie groupoids, smooth natural transformations of smooth functors. ...
Given a projective scheme $X$, say over $\mathbb{C}$, another $\mathbb{C}$-scheme $S$ and a coherent and torsion free (as an $O_X$-module) $M_n(O_X)$-module $F$. Now we can use Morita equivalence to ...