Test *AdjunctMorphismWithGiven

[TKuh]

As requested by @zickgraf here https://github.com/homalg-project/CAP_project/commit/cc441fbbe2ffe9342fc9035dd9b35ad7ab95f578#r138678925

I created new files for these checks (instead of adding them to the existing tests) because they would otherwise be translated to CartesianCategories where they would fail.

Specifically, this test (I'm omitting some details)

hom_da := InternalHomOnObjects( d, a );
hom_da_x_c := TensorProductOnObjects( hom_da, c );

cohom_ad := InternalHomOnObjects( a_op, d_op );
cohom_ad_x_c := TensorProductOnObjects( cohom_ad, c_op );

# Adjoint( Hom( d, a ) → Hom( c, b ) )  ==  Hom( d, a ) ⊗ c → b
hom_to_tensor_adjunction_on_hom_beta_alpha := InternalHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct( c, b, hom_beta_alpha, hom_da_x_c );

# Adjoint( Cohom( b, c ) → Cohom( a, d ) )  ==  b → Cohom( a, d ) ⊗ c
cohom_to_tensor_adjunction_on_cohom_alpha_beta_op := InternalCoHomToTensorProductLeftAdjunctMorphismWithGivenTensorProduct( b_op, c_op, cohom_alpha_beta_op, cohom_ad_x_c );

Assert( 0, IsCongruentForMorphisms( cohom_to_tensor_adjunction_on_cohom_alpha_beta_op, Opposite( opposite, hom_to_tensor_adjunction_on_hom_beta_alpha ) ) );

becomes the following in CartesianCategories:

exp_da := ExponentialOnObjects( d, a );
exp_da_x_c := DirectProduct( exp_da, c );

coexp_ad := ExponentialOnObjects( a_op, d_op );
coexp_ad_x_c := Coproduct( coexp_ad, c_op );

# Adjoint( Exp( d, a ) → Exp( c, b ) )  ==  Exp( d, a ) × c → b
exp_to_tensor_adjunction_on_exp_beta_alpha := ExponentialToDirectProductLeftAdjunctMorphismWithGivenDirectProduct( c, b, exp_beta_alpha, exp_da_x_c );

# Adjoint( Coexp( b, c ) → Coexp( a, d ) )  ==  b → Coexp( a, d ) × c
coexp_to_tensor_adjunction_on_coexp_alpha_beta_op := CoexponentialToCoproductLeftAdjunctMorphismWithGivenCoproduct( b_op, c_op, coexp_alpha_beta_op, coexp_ad_x_c );

# b → Coexp( a, d ) × c  ==  op( Exp( d, a ) × c → b )
Assert( 0, IsCongruentForMorphisms( coexp_to_tensor_adjunction_on_coexp_alpha_beta_op, Opposite( opposite, exp_to_tensor_adjunction_on_exp_beta_alpha ) ) );

The problem is now, that testing with the TerminalCategory lets the above Assert fail:

Assert( 0, IsCongruentForMorphisms( coexp_to_tensor_adjunction_on_coexp_alpha_beta_op, Opposite( opposite, exp_to_tensor_adjunction_on_exp_beta_alpha ) ) );
Error, in function IsCongruentForMorphisms
       of category Opposite with all operations:
       ranges are not equal

gap> r1 := Range(coexp_to_tensor_adjunction_on_coexp_alpha_beta_op);
<A zero object in Opposite with all operations>
gap> r2 := Range( Opposite( opposite, exp_to_tensor_adjunction_on_exp_beta_alpha ) );
<A zero object in Opposite with all operations>

gap> r1 = r2;
false

gap> Display( r1 );
Coproduct

An object in Opposite with all operations given by the above data

gap> Display( r2 );
DirectProduct

An object in Opposite with all operations given by the above data

This is essentially because of the *WithGiven.

However, it works with MonoidalCategories because then both ranges will just be TensorProductOnObjects, hence equal.

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Project coverage is 81.88%. Comparing base (927cc1a) to head (c3f5990).

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[zickgraf]

Thanks @TKuh!

@mohamed-barakat This mainly affects MonoidalCategories and CartesianCategories, will you review/merge?

[mohamed-barakat]

Please rebase and update the version numbers if necessary.

[mohamed-barakat]

Thanks