Generalized Morphism Category by Spans
GAP Categories true or false The GAP category of objects in the generalized morphism category by spans. true or false The GAP category of morphisms in the generalized morphism category by spans.
Properties true or false The argument is a generalized morphism \alpha by a span a \leftarrow b \rightarrow c. The output is true if a \leftarrow b 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 spans. The output is its underlying honest object. a morphism in \mathrm{Hom}_{\mathbf{A}}(b,c) The argument is a generalized morphism \alpha by a span a \leftarrow b \rightarrow c. The output is its arrow b \rightarrow c. a morphism in \mathrm{Hom}_{\mathbf{A}}(b,a) The argument is a generalized morphism \alpha by a span a \leftarrow b \rightarrow c. The output is its reversed arrow a \leftarrow b. a pair of morphisms in \mathbf{A}. The argument is a generalized morphism \alpha: a \rightarrow b by a span. The output is its normalized span pair (a \leftarrow d, d \rightarrow b). a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,a) The argument is a generalized morphism \alpha: a \rightarrow b by a span. 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 span. 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 span 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 span 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 span. The output is its normalization by span.
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 span from the factorobject to the subobject.
Constructors a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,b) The arguments are morphisms \alpha: a \leftarrow c and \beta: c \rightarrow b in \mathbf{A}. The output is a generalized morphism by span with arrow \beta and reversed arrow \alpha. 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 span defined by the composition the given three arrows regarded as generalized morphisms. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(a,c) The arguments are morphisms \alpha: a \rightarrow b, and \beta: b \leftarrow c in \mathbf{A}. The output is a generalized morphism by span defined by the composition 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 span defined by \alpha. a category The argument is an abelian category \mathbf{A}. The output is its generalized morphism category \mathbf{G(A)} by spans. 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 spans whose underlying honest object is a.