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.
l{C}=(l{C}, ⊗,I)
l{Z(C)}
l{C}
uX:A ⊗ X → X ⊗ A
X
uX=(1 ⊗ uY)(uX ⊗ 1)
and
uI=1A
An arrow from (A,u) to (B,v) in
l{Z(C)}
f:A → B
l{C}
vX(f ⊗ 1X)=(1X ⊗ f)uX
lZ(lC)=
End | |
lC ⊗ lCop |
(lC),
i.e., the endofunctors of C which are compatible with the left and right action of C on itself given by the tensor product.
The category
l{Z(C)}
(A,u) ⊗ (B,v)=(A ⊗ B,w)
where
wX=(uX ⊗ 1)(1 ⊗ vX)
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
ModR
ModR
Z(lC):=
End | |
lC ⊗ lCop |
(lC)
Hom(R,R)
The notion of a center in this generality is developed by . Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an
E2
Ek
Ek
Ek+1
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
oplusgVg
In the same vein, 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.
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
Z(A)=
End | |
A ⊗ Aop |
(A).
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.
The categorical trace of a monoidal category (or monoidal ∞-category) is defined as
Tr(C):=C
⊗ | |
C ⊗ Cop |
C.