Description
The type of a Mackey functor homomorphism (for definitions, see background on Mackey functors). A Mackey functor homomorphism can be constructed with an implementation of the map method in this package. Given a Mackey functor homomorphism f, the data for this type can be accessed in the following way:
(Warning: the following method makeRandomMackeyFunctorHomomorphism takes (source,target) as input, which follows the Hom convention rather than the map convention):
i1 : M = makeRandomCpMackeyFunctor(2);
|
i2 : N = makeRandomCpMackeyFunctor(2);
|
i3 : f = makeRandomMackeyFunctorHomomorphism(M,N)
o3 = fix : 0
0 <--------------------- cokernel | 6 |
^ | | 0 |
| | ^ |
| | | |
| v | v
cokernel | 2 | <--------------------- cokernel | 2 |
^ | und : | 0 1 1 0 | | 0 |
└-┘ | 0 |
| 0 |
^ |
└-┘
o3 : MackeyFunctorHomomorphism
|
There are a few ways to build Mackey functor homomorphisms, for example we can take the identity on any Mackey functor
i4 : id_(M)
o4 = fix : | 1 0 |
| 0 1 |
cokernel | 6 | <--------------------- cokernel | 6 |
| 0 | | 0 |
^ | ^ |
| | | |
| v | v
cokernel | 2 | <--------------------- cokernel | 2 |
| 0 | und : | 1 0 0 0 | | 0 |
| 0 | | 0 1 0 0 | | 0 |
| 0 | | 0 0 1 0 | | 0 |
^ | | 0 0 0 1 | ^ |
└-┘ └-┘
o4 : MackeyFunctorHomomorphism
|
The only unary (Boolean-valued) function implemented for Mackey functor homomorphisms is:
The unary, binary, and $n$-ary operations are as follows:
- inversion: if $f$ is an isomorphism we can invert it via f^-1 or using inverse(f)
- negating: we can take the negative of any Mackey functor homomorphism as -f
- equality of two Mackey functors homomorphisms, via f==g
- powers: given an endomorphism of a Mackey functor $f\colon M \to M$ we can take iterated composition of it as f^n
- addition: we can add two Mackey functor homomorphisms with the same domain and codomain via f+g
- subtraction: we can subtract two Mackey functor homomorphisms with the same domain and codomain via f-g
- iterated addition: we can add a Mackey functor homomorphism f with itself n times via n*f
- composition: we can compose two composable Mackey functor homomorphisms via g*f
- block sum: given two Mackey functor homomorphisms $f\colon M_1 \to N_1$ and $g\colon M_2 \to N_2$ we can obtain their block sum $f\oplus g \colon M_1 \oplus M_2 \to N_1 \oplus N_2$ as f++g