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# internalHom -- returns the internal hom Mackey functor between two Mackey functors.

- Usage:
    `internalHom(N,M)`
- Inputs:
  - M, a C_p Mackey Functor,
  - N, a C_p Mackey Functor,
- Outputs:
  - a C_p Mackey Functor, the internal hom Mackey functor from M to N.

## Description

Given any two Mackey functors $M$ and $N$, we can form their internal hom, which is a Mackey functor which we denote by $\underline{\text{Hom}}(M,N)$. For example:

    i1 : internalHom(makeRealRepresentationMackeyFunctor 3, makeComplexRepresentationMackeyFunctor 3)

    o1 =                       Res : | 0 0 -1 |
         image | 0  0  -1 |  --------------------> image | 1 |  -┐ Conj : | 1 |
               | -1 1  -1 | <--------------------        | 1 |  <┘
               | 1  0  1  |    Tr : | 4  |               | 1 |
               | 0  -1 -1 |         | 2  |               | 1 |
               | 1  -1 0  |         | -3 |               | 1 |
               | -1 0  -2 |                              | 1 |
               | 0  1  0  |                              | 2 |
               | 0  0  -1 |                              | 2 |
               | 0  0  -1 |                              | 2 |
               | 0  0  -1 |                              | 1 |
               | 0  0  -2 |                              | 1 |
               | 0  0  -2 |                              | 1 |
               | 0  0  -2 |                              | 1 |
               | 0  0  -1 |                              | 1 |
               | 0  0  -1 |                              | 1 |
               | 0  0  -1 |                              | 1 |
                                                         | 1 |
                                                         | 1 |

    o1 : CpMackeyFunctor

The underlying group of homomorphisms between any two Mackey functors can be recovered as the fixed module key of the internal hom.

## See also

- the abelian category of Mackey functors -- background on the category of Mackey functors and how to code within it
- boxProduct -- box product of Cp-Mackey functors

## Ways to use internalHom:

- `internalHom(CpMackeyFunctor,CpMackeyFunctor)`
- internalHom(CpMackeyFunctor,MackeyFunctorHomomorphism) -- returns the induced map on an internal hom.
- `internalHom(MackeyFunctorHomomorphism,CpMackeyFunctor)` -- see internalHom(CpMackeyFunctor,MackeyFunctorHomomorphism) -- returns the induced map on an internal hom.

## For the programmer

The object internalHom is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CpMackeyFunctors/Documentation/InternalHomDoc.m2:26:0.