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4 Heyting algebras

4.1 Properties

4.1-1 IsHeytingAlgebroid

Returns: true or false

The property of C being a Heyting algebroid.

4.1-2 IsHeytingAlgebra

Returns: true or false

The property of C being a Heyting algebra.

4.2 Operations

4.2-1 NegationOnObjects

Returns: an object

The argument is an object $a$. The output is its negated object $\neg a$.

4.2-2 NegationOnMorphisms

Returns: a morphism in $\mathrm{Hom}( \neg b, \neg a )$.

The argument is a morphism $\alpha: a \rightarrow b$. The output is its negated morphism $\neg \alpha: \neg b \rightarrow \neg a$.

4.2-3 NegationOnMorphismsWithGivenNegations

Returns: a morphism in $\mathrm{Hom}( \neg b, \neg a )$.

The argument is an object $s = \neg b$, a morphism $\alpha: a \rightarrow b$, and an object $r = \neg a$. The output is the negated morphism $\neg \alpha: \neg b \rightarrow \neg a$.

4.2-4 MorphismToDoubleNegation

Returns: a morphism in $\mathrm{Hom}(a, \neg\neg a)$.

The argument is an object $a$. The output is the morphism to the double negation $a \rightarrow \neg\neg a$.

4.2-5 MorphismToDoubleNegationWithGivenDoubleNegation

Returns: a morphism in $\mathrm{Hom}(a, \neg\neg a)$.

The arguments are an object $a$, and an object $r = \neg\neg a$. The output is the morphism to the double negation $a \rightarrow \neg\neg a$.

4.3 Stable internal Hom

4.3-1 StableInternalHom

Returns: a CAP object

Return the stable internal Hom: $\mathrm{\underline{Hom}}(J,\mathrm{\underline{Hom}}(J,...\mathrm{\underline{Hom}}(J,I)...))$.

4.4 Add-methods

4.4-1 AddMorphismToDoubleNegation

Returns: nothing

The arguments are a category $C$ and a function $F$. This operation adds the given function $F$ to the category for the basic operation MorphismToDoubleNegation. Optionally, a weight (default: 100) can be specified which should roughly correspond to the computational complexity of the function (lower weight = less complex = faster execution). $F: ( a ) \mapsto \mathtt{MorphismToDoubleNegation}(a)$.

4.4-2 AddMorphismToDoubleNegationWithGivenDoubleNegation

Returns: nothing

The arguments are a category $C$ and a function $F$. This operation adds the given function $F$ to the category for the basic operation MorphismToDoubleNegationWithGivenDoubleNegation. Optionally, a weight (default: 100) can be specified which should roughly correspond to the computational complexity of the function (lower weight = less complex = faster execution). $F: ( a, r ) \mapsto \mathtt{MorphismToDoubleNegationWithGivenDoubleNegation}(a, r)$.

4.4-3 AddNegationOnMorphisms

Returns: nothing

The arguments are a category $C$ and a function $F$. This operation adds the given function $F$ to the category for the basic operation NegationOnMorphisms. Optionally, a weight (default: 100) can be specified which should roughly correspond to the computational complexity of the function (lower weight = less complex = faster execution). $F: ( alpha ) \mapsto \mathtt{NegationOnMorphisms}(alpha)$.

4.4-4 AddNegationOnMorphismsWithGivenNegations

Returns: nothing

The arguments are a category $C$ and a function $F$. This operation adds the given function $F$ to the category for the basic operation NegationOnMorphismsWithGivenNegations. Optionally, a weight (default: 100) can be specified which should roughly correspond to the computational complexity of the function (lower weight = less complex = faster execution). $F: ( s, alpha, r ) \mapsto \mathtt{NegationOnMorphismsWithGivenNegations}(s, alpha, r)$.

4.4-5 AddNegationOnObjects

Returns: nothing

The arguments are a category $C$ and a function $F$. This operation adds the given function $F$ to the category for the basic operation NegationOnObjects. Optionally, a weight (default: 100) can be specified which should roughly correspond to the computational complexity of the function (lower weight = less complex = faster execution). $F: ( arg2 ) \mapsto \mathtt{NegationOnObjects}(arg2)$.

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