‣ 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).
‣ 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).
‣ 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).
‣ 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).
‣ 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 \(\delta \circ \mu_2\).
‣ 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 \(\delta \circ \mu_2\).
‣ 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 \(\rho_S \circ \delta\).
‣ 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 \(\rho_S \circ \delta\).
‣ 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]).
‣ 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).
‣ 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.
11.3-1 \*‣ \*( 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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