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BiHom Hopf algebras viewed as Hopf monoids
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We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them
L
a
x
+
O
p
l
a
x
0
\mathsf {Lax}^+\mathsf {Oplax}^0
-monoidal. Dually, we consider
L
a
x
0
O
p
l
a
x
+
\mathsf {Lax}_0\mathsf {Oplax}_+
-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define
L
a
x
0
+
O
p
l
a
x
+
0
\mathsf {Lax}^+_0\mathsf {Oplax}^0_+
-duoidal categories with compatible
L
a
x
+
O
p
l
a
x
0
\mathsf {Lax}^+\mathsf {Oplax}^0
- and
L
a
x
0
O
p
l
a
x
+
\mathsf {Lax}_0\mathsf {Oplax}_+
-monoidal structures. We introduce comonoids in
L
a
x
+
O
p
l
a
x
0
\mathsf {Lax}^+\mathsf {Oplax}^0
-monoidal categories, monoids in
L
a
x
0
O
p
l
a
x
+
\mathsf {Lax}_0\mathsf {Oplax}_+
-monoidal categories and bimonoids in
L
a
x
0
+
O
p
l
a
x
+
0
\mathsf {Lax}^+_0\mathsf {Oplax}^0_+
-duoidal categories.
Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a
L
a
x
0
+
O
p
l
a
x
+
0
\mathsf {Lax}^+_0\mathsf {Oplax}^0_+
-duoidal category
D
\mathsf D
to any symmetric monoidal category
V
\mathsf V
. The unital
B
i
H
o
m
\mathsf {BiHom}
-monoids, counital
B
i
H
o
m
\mathsf {BiHom}
-comonoids, and unital and counital
B
i
H
o
m
\mathsf {BiHom}
-bimonoids in
V
\mathsf V
, due to Grazianu et al., are identified with the monoids, comonoids and bimonoids in
D
\mathsf D
, respectively.
Title: BiHom Hopf algebras viewed as Hopf monoids
Description:
We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent.
We call them
L
a
x
+
O
p
l
a
x
0
\mathsf {Lax}^+\mathsf {Oplax}^0
-monoidal.
Dually, we consider
L
a
x
0
O
p
l
a
x
+
\mathsf {Lax}_0\mathsf {Oplax}_+
-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products.
We define
L
a
x
0
+
O
p
l
a
x
+
0
\mathsf {Lax}^+_0\mathsf {Oplax}^0_+
-duoidal categories with compatible
L
a
x
+
O
p
l
a
x
0
\mathsf {Lax}^+\mathsf {Oplax}^0
- and
L
a
x
0
O
p
l
a
x
+
\mathsf {Lax}_0\mathsf {Oplax}_+
-monoidal structures.
We introduce comonoids in
L
a
x
+
O
p
l
a
x
0
\mathsf {Lax}^+\mathsf {Oplax}^0
-monoidal categories, monoids in
L
a
x
0
O
p
l
a
x
+
\mathsf {Lax}_0\mathsf {Oplax}_+
-monoidal categories and bimonoids in
L
a
x
0
+
O
p
l
a
x
+
0
\mathsf {Lax}^+_0\mathsf {Oplax}^0_+
-duoidal categories.
Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts.
This assigns a
L
a
x
0
+
O
p
l
a
x
+
0
\mathsf {Lax}^+_0\mathsf {Oplax}^0_+
-duoidal category
D
\mathsf D
to any symmetric monoidal category
V
\mathsf V
.
The unital
B
i
H
o
m
\mathsf {BiHom}
-monoids, counital
B
i
H
o
m
\mathsf {BiHom}
-comonoids, and unital and counital
B
i
H
o
m
\mathsf {BiHom}
-bimonoids in
V
\mathsf V
, due to Grazianu et al.
, are identified with the monoids, comonoids and bimonoids in
D
\mathsf D
, respectively.
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