In algebra, the Yoneda product is the pairing between Ext groups of modules:
Ext
n
(
M
,
N
)
⊗
Ext
m
(
L
,
M
)
→
Ext
n
+
m
(
L
,
N
)
induced by
Hom
(
M
,
N
)
⊗
Hom
(
L
,
M
)
→
Hom
(
L
,
N
)
,
f
⊗
g
↦
f
∘
g
.
Specifically, for an element
ξ
∈
Ext
n
(
M
,
N
)
, thought of as an extension
ξ
:
0
→
N
→
E
0
→
⋯
→
E
n
−
1
→
M
→
0
,
and similarly
ρ
:
0
→
M
→
F
0
→
⋯
→
F
m
−
1
→
L
→
0
∈
Ext
m
(
L
,
M
)
,
we form the Yoneda (cup) product
ξ
⌣
ρ
:
0
→
N
→
E
0
→
⋯
→
E
n
−
1
→
F
0
→
⋯
→
F
m
−
1
→
L
→
0
∈
Ext
m
+
n
(
L
,
N
)
.
Note that the middle map
E
n
−
1
→
F
1
factors through the given maps to
M
.
We extend this definition to include
m
,
n
=
0
using the usual functoriality of the
Ext
∗
(
_
,
_
)
groups.
