• Source: Center (category theory)

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





C


=
(


C


,
⊗
,
I
)


{\displaystyle {\mathcal {C}}=({\mathcal {C}},\otimes ,I)}

, denoted





Z
(
C
)




{\displaystyle {\mathcal {Z(C)}}}

, is the category whose objects are pairs (A,u) consisting of an object A of





C




{\displaystyle {\mathcal {C}}}

and an isomorphism




u

X


:
A
⊗
X
→
X
⊗
A


{\displaystyle u_{X}:A\otimes X\rightarrow X\otimes A}

which is natural in



X


{\displaystyle X}

satisfying





u

X
⊗
Y


=
(
1
⊗

u

Y


)
(

u

X


⊗
1
)


{\displaystyle u_{X\otimes Y}=(1\otimes u_{Y})(u_{X}\otimes 1)}


and





u

I


=

1

A




{\displaystyle u_{I}=1_{A}}

(this is actually a consequence of the first axiom).
An arrow from (A,u) to (B,v) in





Z
(
C
)




{\displaystyle {\mathcal {Z(C)}}}

consists of an arrow



f
:
A
→
B


{\displaystyle f:A\rightarrow B}

in





C




{\displaystyle {\mathcal {C}}}

such that





v

X


(
f
⊗

1

X


)
=
(

1

X


⊗
f
)

u

X




{\displaystyle v_{X}(f\otimes 1_{X})=(1_{X}\otimes f)u_{X}}

.
This definition of the center appears in Joyal & Street (1991). Equivalently, the center may be defined as






Z


(


C


)
=


E
n
d




C


⊗



C



o
p




(


C


)
,


{\displaystyle {\mathcal {Z}}({\mathcal {C}})=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}}),}


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





Z
(
C
)




{\displaystyle {\mathcal {Z(C)}}}

becomes a braided monoidal category with the tensor product on objects defined as




(
A
,
u
)
⊗
(
B
,
v
)
=
(
A
⊗
B
,
w
)


{\displaystyle (A,u)\otimes (B,v)=(A\otimes B,w)}


where




w

X


=
(

u

X


⊗
1
)
(
1
⊗

v

X


)


{\displaystyle w_{X}=(u_{X}\otimes 1)(1\otimes v_{X})}

, 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





M
o
d


R




{\displaystyle \mathrm {Mod} _{R}}

of R-modules, for a commutative ring R, is





M
o
d


R




{\displaystyle \mathrm {Mod} _{R}}

again. The center of a monoidal ∞-category C can be defined, analogously to the above, as




Z
(


C


)
:=


E
n
d




C


⊗



C



o
p




(


C


)


{\displaystyle Z({\mathcal {C}}):=\mathrm {End} _{{\mathcal {C}}\otimes {\mathcal {C}}^{op}}({\mathcal {C}})}

.
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



H
o
m
(
R
,
R
)


{\displaystyle Hom(R,R)}

(derived Hom).
The notion of a center in this generality is developed by Lurie (2017, §5.3.1). Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an




E

2




{\displaystyle E_{2}}

-monoidal category. More generally, the center of a




E

k




{\displaystyle E_{k}}

-monoidal category is an algebra object in




E

k




{\displaystyle E_{k}}

-monoidal categories and therefore, by Dunn additivity, an




E

k
+
1




{\displaystyle E_{k+1}}

-monoidal category.




Examples


Hinich (2007) 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





⨁

g
∈
G



V

g




{\displaystyle \bigoplus _{g\in G}V_{g}}


for some k-vector spaces, together with G-equivariant morphisms, where G acts on itself by conjugation.
In the same vein, Ben-Zvi, Francis & Nadler (2010) 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




Z
(
A
)
=
E
n

d

A
⊗

A

o
p




(
A
)
.


{\displaystyle Z(A)=End_{A\otimes A^{op}}(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.




= Categorical trace

=
The categorical trace of a monoidal category (or monoidal ∞-category) is defined as




T
r
(
C
)
:=
C

⊗

C
⊗

C

o
p




C
.


{\displaystyle Tr(C):=C\otimes _{C\otimes C^{op}}C.}


The concept is being widely applied, for example in Zhu (2018).




References






External links


Drinfeld center at the nLab

Kata Kunci Pencarian:

• Agama di Tiongkok
• Agama
• Cody Rhodes
• Kanae Yamamoto
• Kriptografi
• Vaksin
• Homoseksualitas
• Perbudakan
• Boron
• Scandal
• Center (category theory)
• Center (algebra)
• Representation theory
• Center (group theory)
• Prototype theory
• GNS theory
• John C. Baez
• William Lawvere
• Equivariant map
• Horseshoe theory