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### 11 Freyd category

#### 11.1 Random methods in Freyd categories

##### 11.1-1 RandomObjectByList
 ‣ RandomObjectByList( cat, L ) ( operation )

Returns: an object in a Freyd category

The arguments are a Freyd category cat of a category underlying_cat and a list L. The output is an object in cat whose relation morphism is a random morphism in underlying_cat constructed via RandomMorphismByList(underlying_cat, L).

##### 11.1-2 RandomObjectByInteger
 ‣ RandomObjectByInteger( cat, n ) ( operation )

Returns: an object in a Freyd category

The arguments are a Freyd category cat of a category underlying_cat and an integer n. The output is an object in cat whose relation morphism is a random morphism in underlying_cat constructed via RandomMorphismByInteger(underlying_cat, n).

##### 11.1-3 RandomMorphismWithFixedSourceAndRangeByList
 ‣ RandomMorphismWithFixedSourceAndRangeByList( S, R, L ) ( operation )

Returns: a morphism in a Freyd category

The arguments are two objects S and R in a Freyd category cat of a category underlying_cat and a list L. The category cat is required to have a homomorphism structure (1,H,\nu) over underlying_cat or cat. The output is a morphism in cat constructed as follows:

• If the source of the relation morphism \rho_S of S is zero, then the morphism datum of the output is constructed via RandomMorphismWithFixedSourceAndRangeByList(Range(\rho_S), Range(\rho_R), L) where \rho_R is the relation morphism of R.

• Otherwise, the output is \nu^{-1}_{S,R}(\eta):S \to R where \eta:1\to H(S,R) is a random morphism constructed via RandomMorphismWithFixedSourceAndRangeByList(1,H(S,R),L).

##### 11.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
 ‣ RandomMorphismWithFixedSourceAndRangeByInteger( S, R, n ) ( operation )

Returns: a morphism in a Freyd category

The arguments are two objects S and R in a Freyd category cat of a category underlying_cat and an integer n. The category cat is required to have a homomorphism structure (1,H,\nu) over underlying_cat or cat. The output is a morphism in cat constructed as follows:

• If the source of the relation morphism \rho_S of S is zero, then the morphism datum of the output is constructed via RandomMorphismWithFixedSourceAndRangeByInteger(Range(\rho_S), Range(\rho_R), n) where \rho_R is the relation morphism of R.

• Otherwise, the output is \nu^{-1}_{S,R}(\eta):S \to R where \eta:1\to H(S,R) is a random morphism constructed via RandomMorphismWithFixedSourceAndRangeByInteger(1,H(S,R),n).

##### 11.1-5 RandomMorphismWithFixedSourceByList
 ‣ RandomMorphismWithFixedSourceByList( S, L ) ( operation )

Returns: a morphism in a Freyd category

The arguments are an object S in a Freyd category cat of a category underlying_cat and a list L. We denote the relation morphism of S by \rho_S:S_1 \to S_2 in underlying_cat. The output is a morphism in cat constructed as follows:

• Compute a morphism \rho:S_1 \to R_1 via RandomMorphismWithFixedSourceByList(S_1,L).

• Compute the pair of injections \mu_1:S_2 \to K and \mu_2:R_1 \to K in the weak-bipushout K of the pair (\rho_S, \rho).

• Compute a morphism \delta:K \to R_2 via RandomMorphismWithFixedSourceByList(K,L).

The output is a morphism S \to R in cat whose morphism datum is \delta \circ \mu_1 and whose range's relation morphism is \delta \circ \mu_2.

##### 11.1-6 RandomMorphismWithFixedSourceByInteger
 ‣ RandomMorphismWithFixedSourceByInteger( S, n ) ( operation )

Returns: a morphism in a Freyd category

The arguments are an object S in a Freyd category cat of a category underlying_cat and an integer n. We denote the relation morphism of S by \rho_S:S_1 \to S_2 in underlying_cat. The output is a morphism in cat constructed as follows:

• Compute a morphism \rho:S_1 \to R_1 via RandomMorphismWithFixedSourceByInteger(S_1,n).

• Compute the pair of injections \mu_1:S_2 \to K and \mu_2:R_1 \to K in the weak-bipushout K of the pair (\rho_S, \rho).

• Compute a morphism \delta:K \to R_2 via RandomMorphismWithFixedSourceByInteger(K,n).

The output is a morphism S \to R in cat whose morphism datum is \delta \circ \mu_1 and whose range's relation morphism is \delta \circ \mu_2.

##### 11.1-7 RandomMorphismWithFixedRangeByList
 ‣ RandomMorphismWithFixedRangeByList( R, L ) ( operation )

Returns: a morphism in a Freyd category

The arguments are an object R in a Freyd category cat of a category underlying_cat and a list L. We denote the relation morphism of R by \rho_R:R_1 \to R_2 in underlying_cat. The output is a morphism in cat constructed as follows:

• Compute a morphism \rho:S_2 \to R_2 via RandomMorphismWithFixedRangeByList(R_2,L).

• Compute the projection \rho_S:K \to S_2 in the second factor of the weak-bifiber-product of the pair (\rho_R,\rho).

• Compute a morphism \delta:S_1 \to K via RandomMorphismWithFixedRangeByList(K,L).

The output is a morphism S \to R in cat whose morphism datum is \rho and whose source's relation morphism is \rho_S \circ \delta.

##### 11.1-8 RandomMorphismWithFixedRangeByInteger
 ‣ RandomMorphismWithFixedRangeByInteger( R, n ) ( operation )

Returns: a morphism in a Freyd category

The arguments are an object R in a Freyd category cat of a category underlying_cat and an integer n. We denote the relation morphism of R by \rho_R:R_1 \to R_2 in underlying_cat. The output is a morphism in cat constructed as follows:

• Compute a morphism \rho:S_2 \to R_2 via RandomMorphismWithFixedRangeByInteger(R_2,n).

• Compute the projection \rho_S:K \to S_2 in the second factor of the weak-bifiber-product of the pair (\rho_R,\rho).

• Compute a morphism \delta:S_1 \to K via RandomMorphismWithFixedRangeByInteger(K,n).

The output is a morphism S \to R in cat whose morphism datum is \rho and whose source's relation morphism is \rho_S \circ \delta.

##### 11.1-9 RandomMorphismByList
 ‣ RandomMorphismByList( cat, L ) ( operation )

Returns: a morphism in a Freyd category

The arguments are a Freyd category cat and a list L consisting of three lists. The output is constructed via RandomMorphismWithFixedSourceAndRangeByList(S,R,L[3]) where S is constructed via RandomObjectByList(cat,L[1]) and R via RandomObjectByList(cat,L[2]).

##### 11.1-10 RandomMorphismByInteger
 ‣ RandomMorphismByInteger( cat, n ) ( operation )

Returns: a morphism in a Freyd category

The arguments are a Freyd category cat and an integer n. The output is constructed via RandomMorphismWithFixedSourceAndRangeByInteger(S,R,n) where S and R are random objects constructed via RandomObjectByInteger(cat,n).

#### 11.2 Internal Hom-Embedding

##### 11.2-1 INTERNAL_HOM_EMBEDDING
 ‣ INTERNAL_HOM_EMBEDDING( cat, a, b ) ( operation )

Returns: a (mono)morphism

The arguments are two objects a and b of a Freyd category cat. Assume that the relation morphism for a is \alpha \colon R_A \to A, then we have the exact sequence 0 \to \mathrm{\underline{Hom}} \left( a,b \right) \to \mathrm{\underline{Hom}}(A, b) \to \mathrm{\underline{Hom}}(R_A, b). The embedding of \mathrm{\underline{Hom}}( a, b ) into \mathrm{\underline{Hom}}(A, b) is the internal Hom-embedding. This method returns this very map.

#### 11.3 Convenient methods for tensor products of freyd objects and morphisms

##### 11.3-1 \*
 ‣ \*( arg1, arg2 ) ( operation )

##### 11.3-2 \^
 ‣ \^( arg1, arg2 ) ( operation )

##### 11.3-3 \*
 ‣ \*( arg1, arg2 ) ( operation )

##### 11.3-4 \^
 ‣ \^( arg1, arg2 ) ( operation )
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