29
votes
Why is the bicategory viewpoint useful?
You believe the 1-category is interesting but somehow not the bicategory, yet one could say the exact same thing one dimension below: Why is this category even interesing while you could just consider ...
- 42.4k
20
votes
Accepted
Does Morita theory hint higher modules for noncommutative ring?
Yes. The trick is to use not just categories, but pointed categories, which are categories equipped with a choice of object (the "pointing"). Given any ring $R$, the category $\mathrm{Mod}(R)...
- 54.6k
12
votes
Accepted
Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?
In the non-unitary setting ENO proved that if $Z(C)$ and $Z(D)$ are equivalent as braided tensor categories, then C and D are Morita equivalent. This is Theorem 3.1 of this paper. Note that they ...
- 28.1k
11
votes
Accepted
Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?
I don't know of any "clever" name, but there are names that appear in the literature, such as the tensor product of Grothendieck categories, or the category of (additive) locally presentable ...
- 53.3k
11
votes
Accepted
Is there a strongly noncommutative fusion category?
Consider the symmetric group group $G = S_3$ of order $6$. Then $\mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3_{\mathrm{gp}}(G;\...
- 54.6k
9
votes
Condition for an equivalence of functor categories to imply an equivalence of categories
You can find this theoerm as Proposition 5.28 in Kelly's Basic concepts of enriched category theory. I guess the only extra condition you need is that the base for enrichment forms a topos. They are ...
- 1,482
8
votes
Differing notions of Morita equivalence for operator algebras
I will have to look for a reference, but strong Morita equivalence is the notion of equivalence in the $\rm C^*$-2-category of $\rm C^*$-algebras, Hilbert $\rm C^*$-correspondences, and adjointable ...
- 5,425
8
votes
Accepted
Picard-surjectivity and Morita-equivalence
Yes, the basic algebra of $A$ will be Picard-surjective.
The basic algebra is the endomorphism algebra $\operatorname{End}_A(\bigoplus_{i=1}^{n}P_i)$ of the direct sum of indecomposable projective (...
- 35.2k
8
votes
Accepted
Morita equivalences and centers of some algebras
The answer is that in your matrix $\left(\begin{smallmatrix} 0&x_0\\0&0\end{smallmatrix}\right)$, the $x_0$ denotes the isomorphism of modules given by left multiplication by $x_0$, so it ...
- 16.2k
8
votes
Condition for an equivalence of functor categories to imply an equivalence of categories
As AT0 said, there is an analogous theorem in Kelly's book for categories enriched over any base. So this will apply as soon as $C$ and $D$ are enriched over $\mathcal{S}$. But if $\mathcal{S}$ is a ...
- 66.7k
8
votes
Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?
I say “Deligne-Kelly tensor product” for this. Technically Deligne tensor product is for certain abelian categories and right exact functors, and Kelly is for finitely cocomplete categories and right ...
- 28.1k
7
votes
Morita-invertible C*-algebras
I know my answer is coming a bit late, but the answer to your question is: yes. If $A$ is a $C^\ast$-algebra, and there exists a $C^\ast$-algebra $B$ such that $A\otimes_\alpha B$ is strongly Morita ...
- 2,471
7
votes
Is it true that $A$ is Morita equivalent with $M_I(A)$
Taking $A= \mathbb{C}$ , $A$ is not Morita equivalent to $B=M_I (A)$ when $I$ is infinite, since $A$ is artinian but $M_I (A)$ is not (here I assume that $M_I(A)$ is defined as the endomorphism ring ...
- 26.5k
7
votes
Is a Morita equivalent functor an exact functor(Module protective direct sum) ?
Let $A=kQ$ be the path algebra of Dynkin type $\mathcal{A_2}$ and $B=K[x]/(x^2)$. Both algebras have a unique simple non-projective module and all other indecomposable modules are projective. Thus ...
- 26.5k
6
votes
Accepted
Morita equivalence and connectivity
Suppose $A$ is any nontrivial connective ring spectrum, with right module $P = A \oplus A[1]$, and let $B = End_A(P)$.
The ring $B$ satisfies
$$B \simeq A \oplus A[1] \oplus A[-1] \oplus A$$
and in ...
- 52.6k
6
votes
Accepted
Morita equivalence and isomorphisms in cohomology theories
The conceptual point is that all of these invariants are Morita invariant because they can be defined directly in terms of the category of modules. Explicitly:
Starting from the category of modules $\...
- 118k
5
votes
Accepted
Dirac operator on a Morita equivalent algebra
Your question is entirely covered by Section 2 of Brain–Mesland–Van Suijlekom, but the fgp case is simple enough to ultimately boil down to folklore proved by Chakraborty–Mathai. Let me summarise what ...
- 2,385
5
votes
Accepted
Morita equivalence of quivers from related exceptional collections
The algebras $B_1$ and $B_2$ are NOT Morita-equivalent. Indeed, their categories of modules have exactly three simple (i.e., without non-trivial subobjects) objects, that correspond to the vertices of ...
- 39.3k
5
votes
Accepted
Morita equivalence of Lie groupoids and isomorphism of differentiable stacks
The "well-known fact" is simply not true if you assume "isomorphic stacks" means literally isomorphic (say as fibred categories). My impression is that people who work in certain ...
- 35.4k
4
votes
Categorical Morita equivalence implies equivalence of module categories?
Yes, it is Theorem 7.12.16 in [1]. In fact these are 2-equivalent (as the categories of modules are 2-categories).
Theorem 7.12.16. Let $M$ be a faithful exact module category over $C$. The $2$-...
- 639
4
votes
Isomorphism classes of rings of differential operators
The paper
J. Grabowski: Isomorphisms and Ideals of the Lie Algebras of Vector Fields, Inventiones Math. 50, 13-33 (1978)
shows that any smooth manifold (or real analytic manifold, or Stein manifold)...
- 25.3k
3
votes
Accepted
Whether Morita equivalence holds the following properties?
1) Yes, Morita equivalence trivially implies derived equivalence. Note that two algebras over an algebraically closed field are Morita equivalent iff their quiver algebras are isomorphic. So compared ...
- 26.5k
3
votes
Accepted
Necessity/Motivation for generalised homomorpisms
Let me attempt a very simple-minded answer.
Say your objects of interest are orbifolds. You have an orbifold V and you want to describe it through a groupoid, usually an action groupoid $G\ltimes M \...
- 3,377
3
votes
Does Morita theory hint higher modules for noncommutative ring?
Although I think my answer "pointed categories" is an important one, there is another way that the question could be interpreted: What is an interesting class of rings which are recoverable ...
- 54.6k
2
votes
Differing notions of Morita equivalence for operator algebras
Given a $C^*$-algebra $C$, let us write $\operatorname{Mod}(C)$ for the category (without identities!) with as objects right Hilbert $C^*$-modules and as morphisms adjointable compact operators.
...
- 525
2
votes
Accepted
What to call a morphism of sites inducing an equivalence on categories of sheaves?
Johnstone's Sketches of an Elephant (2 volumes) is a standard reference which uses "Morita equivalence" in this way. In fact, Jonstone systematically uses "Morita equivalence" in a similar way across ...
- 63.9k
2
votes
Necessity/Motivation for generalised homomorpisms
When one studies equivariant geometry, then it is sometimes the case that one can pass to a smaller space with the action of a smaller group, and everything equivariant you can compute will be the ...
- 35.4k
2
votes
Morita equivalence of $K$-algebras
If V is a vector space such that $V\oplus V$ is isomorphic to V then A=TV, the tensor álgebra, is isomorphic to $T(V\oplus V)=A\coprod A$, in particular Morita equivalent
- 1,028
2
votes
Morita equivalence of DG algebras? (reference needed)
Just a comment on "For example, every linear invariant that I know of (algebraic K-theory, cyclic (co)homology, Hochschild (co)homology,...) is not only invariant under Morita equivalence but also ...
- 1,028
1
vote
Morita equivalent algebras in a fusion category
in the algebra case, B=eMn(A)e "because" B=End_A (P) with P f.g. proyective, so, finding e is the same as give a presentation of P as a direct summand of A^n. Also, P=F(B) where F is the functor ...
- 1,028
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