Generalized Morphism Category by Cospans
GAP Categories true or false The GAP category of objects in the generalized morphism category by cospans. true or false The GAP category of morphisms in the generalized morphism category by cospans.
Properties true or false The argument is a generalized morphism \alpha by a cospan a \rightarrow b \leftarrow c. The output is true if b \leftarrow c is congruent to an identity morphism, false otherwise.
Attributes an object in \mathbf{A} The argument is an object a in the generalized morphism category by cospans. The output is its underlying honest object. a morphism in \mathrm{Hom}_{\mathbf{A}}(a,c) The argument is a generalized morphism \alpha by a cospan a \rightarrow b \leftarrow c. The output is its arrow a \rightarrow b. a morphism in \mathrm{Hom}_{\mathbf{A}}(c,b) The argument is a generalized morphism \alpha by a cospan a \rightarrow b \leftarrow c. The output is its reversed arrow b \leftarrow c. a pair of morphisms in \mathbf{A}. The argument is a generalized morphism \alpha: a \rightarrow b by a cospan. The output is its normalized cospan pair (a \rightarrow d, d \leftarrow b). a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,a) The argument is a generalized morphism \alpha: a \rightarrow b by a cospan. The output is its pseudo inverse b \rightarrow a. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,a) The argument is a morphism \alpha: a \rightarrow b \in \mathbf{A}. The output is its generalized inverse b \rightarrow a by cospan. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,b) The argument is a subobject \alpha: a \hookrightarrow b \in \mathbf{A}. The output is the idempotent b \rightarrow b \in \mathbf{G(A)} by cospan defined by \alpha. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,b) The argument is a factorobject \alpha: b \twoheadrightarrow a \in \mathbf{A}. The output is the idempotent b \rightarrow b \in \mathbf{G(A)} by cospan defined by \alpha. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,b) The argument is a generalized morphism \alpha: a \rightarrow b by a cospan. The output is its normalization by cospan.
Operations a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(c,a) The arguments are a a factorobject \beta: b \twoheadrightarrow c, and a subobject \alpha: a \hookrightarrow b. The output is the generalized morphism by cospan from the factorobject to the subobject.
Constructors a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,c) The arguments are morphisms \alpha: a \rightarrow b and \beta: c \rightarrow b in \mathbf{A}. The output is a generalized morphism by cospan with arrow \alpha and reversed arrow \beta. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,d) The arguments are morphisms \alpha: a \leftarrow b, \beta: b \rightarrow c, and \gamma: c \leftarrow d in \mathbf{A}. The output is a generalized morphism by cospan defined by the composition of the given three arrows regarded as generalized morphisms. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,c) The arguments are morphisms \alpha: a \leftarrow b, and \beta: b \rightarrow c in \mathbf{A}. The output is a generalized morphism by cospan defined by the composition of the given two arrows regarded as generalized morphisms. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,b) The argument is a morphism \alpha: a \rightarrow b in \mathbf{A}. The output is the honest generalized morphism by cospan defined by \alpha. a category The argument is an abelian category \mathbf{A}. The output is its generalized morphism category \mathbf{G(A)} by cospans. an object in \mathbf{G(A)} The argument is an object a in an abelian category \mathbf{A}. The output is the object in the generalized morphism category by cospans whose underlying honest object is a.
Constructors of lifts of exact functors and natrual (iso)morphisms Lift the exact functor F to a functor A \to B, where A := GeneralizedMorphismCategoryByCospans( AsCapCategory( Source( F ) ) ) and B := GeneralizedMorphismCategoryByCospans( AsCapCategory( Range( F ) ) ).