Generalized Morphism Category by Three Arrows
GAP Categories true or false The GAP category of objects in the generalized morphism category by three arrows. true or false The GAP category of morphisms in the generalized morphism category by three arrows.
Properties true or false The argument is a generalized morphism \alpha by three arrows a \leftarrow b \rightarrow c \leftarrow d. The output is true if a \leftarrow b and c \leftarrow d are congruent to identity morphisms, false otherwise. true or false The argument is a generalized morphism \alpha by three arrows a \leftarrow b \rightarrow c \leftarrow d. The output is true if a \leftarrow b is congruent to an identity morphism, false otherwise. true or false The argument is a generalized morphism \alpha by three arrows a \leftarrow b \rightarrow c \leftarrow d. The output is true if c \leftarrow d 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 three arrows. The output is its underlying honest object. a morphism in \mathrm{Hom}_{\mathbf{A}}(b,a) The argument is a generalized morphism \alpha by three arrows a \leftarrow b \rightarrow c \leftarrow d. The output is its source aid a \leftarrow b. a morphism in \mathrm{Hom}_{\mathbf{A}}(d,c) The argument is a generalized morphism \alpha by three arrows a \leftarrow b \rightarrow c \leftarrow d. The output is its range aid c \leftarrow d. a morphism in \mathrm{Hom}_{\mathbf{A}}(b,c) The argument is a generalized morphism \alpha by three arrows a \leftarrow b \rightarrow c \leftarrow d. The output is its range aid b \rightarrow c. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,a) The argument is a generalized morphism \alpha: a \rightarrow b by three arrows. 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 three arrows. 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 three arrows 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 three arrows defined by \alpha.
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 three arrows from the factorobject to the subobject. a list of generalized morphisms The argument is a list L of generalized morphisms by three arrows having the same range. The output is a list of generalized morphisms by three arrows which is the comman coastriction of L.
Constructors 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 three arrows with source aid \alpha, arrow \beta, and range aid \gamma. 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 three arrows defined by the composition of the given two arrows regarded as generalized morphisms. a morphism in \mathrm{Hom}_{\mathbf{G(A)}}(b,d) The arguments are morphisms \beta: b \rightarrow c, and \gamma: c \leftarrow d in \mathbf{A}. The output is a generalized morphism by three arrows 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 three arrows defined by \alpha. a category The argument is an abelian category \mathbf{A}. The output is its generalized morphism category \mathbf{G(A)} by three arrows. 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 three arrows whose underlying honest object is a.