Английская Википедия:Center (category theory)
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In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the notion of the center of a monoid, group, or ring to a category.
Definition
The center of a monoidal category <math>\mathcal{C} = (\mathcal{C},\otimes,I)</math>, denoted <math>\mathcal{Z(C)}</math>, is the category whose objects are pairs (A,u) consisting of an object A of <math>\mathcal{C}</math> and an isomorphism <math>u_X:A \otimes X \rightarrow X \otimes A</math> which is natural in <math>X</math> satisfying
- <math>u_{X \otimes Y} = (1 \otimes u_Y)(u_X \otimes 1)</math>
and
- <math>u_I = 1_A</math> (this is actually a consequence of the first axiom).Шаблон:Sfn
An arrow from (A,u) to (B,v) in <math>\mathcal{Z(C)}</math> consists of an arrow <math>f:A \rightarrow B</math> in <math>\mathcal{C}</math> such that
- <math>v_X (f \otimes 1_X) = (1_X \otimes f) u_X</math>.
This definition of the center appears in Шаблон:Harvtxt. Equivalently, the center may be defined as
- <math>\mathcal Z(\mathcal C) = \mathrm{End}_{\mathcal C \otimes \mathcal C^{op}}(\mathcal C),</math>
i.e., the endofunctors of C which are compatible with the left and right action of C on itself given by the tensor product.
Braiding
The category <math>\mathcal{Z(C)}</math> becomes a braided monoidal category with the tensor product on objects defined as
- <math>(A,u) \otimes (B,v) = (A \otimes B,w)</math>
where <math>w_X = (u_X \otimes 1)(1 \otimes v_X)</math>, and the obvious braiding.
Higher categorical version
The categorical center is particularly useful in the context of higher categories. This is illustrated by the following example: the center of the (abelian) category <math>\mathrm{Mod}_R</math> of R-modules, for a commutative ring R, is <math>\mathrm{Mod}_R</math> again. The center of a monoidal ∞-category C can be defined, analogously to the above, as
- <math>Z(\mathcal C) := \mathrm{End}_{\mathcal C \otimes \mathcal C^{op}}(\mathcal C)</math>.
Now, in contrast to the above, the center of the derived category of R-modules (regarded as an ∞-category) is given by the derived category of modules over the cochain complex encoding the Hochschild cohomology, a complex whose degree 0 term is R (as in the abelian situation above), but includes higher terms such as <math>Hom(R, R)</math> (derived Hom).[1]
The notion of a center in this generality is developed by Шаблон:Harvtxt. Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an <math>E_2</math>-monoidal category. More generally, the center of a <math>E_k</math>-monoidal category is an algebra object in <math>E_k</math>-monoidal categories and therefore, by Dunn additivity, an <math>E_{k+1}</math>-monoidal category.
Examples
Шаблон:Harvtxt has shown that the Drinfeld center of the category of sheaves on an orbifold X is the category of sheaves on the inertia orbifold of X. For X being the classifying space of a finite group G, the inertia orbifold is the stack quotient G/G, where G acts on itself by conjugation. For this special case, Hinich's result specializes to the assertion that the center of the category of G-representations (with respect to some ground field k) is equivalent to the category consisting of G-graded k-vector spaces, i.e., objects of the form
- <math>\bigoplus_{g \in G} V_g</math>
for some k-vector spaces, together with G-equivariant morphisms, where G acts on itself by conjugation.
In the same vein, Шаблон:Harvtxt have shown that Drinfeld center of the derived category of quasi-coherent sheaves on a perfect stack X is the derived category of sheaves on the loop stack of X.
Related notions
Centers of monoid objects
The center of a monoid and the Drinfeld center of a monoidal category are both instances of the following more general concept. Given a monoidal category C and a monoid object A in C, the center of A is defined as
- <math>Z(A) = End_{A \otimes A^{op}}(A).</math>
For C being the category of sets (with the usual cartesian product), a monoid object is simply a monoid, and Z(A) is the center of the monoid. Similarly, if C is the category of abelian groups, monoid objects are rings, and the above recovers the center of a ring. Finally, if C is the category of categories, with the product as the monoidal operation, monoid objects in C are monoidal categories, and the above recovers the Drinfeld center.
Categorical trace
The categorical trace of a monoidal category (or monoidal ∞-category) is defined as
- <math>Tr(C) := C \otimes_{C \otimes C^{op}} C.</math>
The concept is being widely applied, for example in Шаблон:Harvtxt.
References
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