CAP-2019.06.07/gap/CategoryMorphisms.gd

12 #!  After a GAP object is added to the category, it knows which things can be
27 #! The argument is a morphism $\alpha$.
28 #! The output is the category $\mathbf{C}$
36 #! The argument is a morphism $\alpha: a \rightarrow b$.
37 #! The output is its source $a$.
44 #! The argument is a morphism $\alpha: a \rightarrow b$.
45 #! The output is its range $b$.
51 # this attribute is also an implied operation
76 # The argument is a morphism $\alpha$.
77 # The output is <C>true</C> if $\alpha$ is a monomorphism,
78 # otherwise the output is <C>false</C>.
109 # The argument is a morphism $\alpha$.
110 # The output is <C>true</C> if $\alpha$ is an epimorphism,
111 # otherwise the output is <C>false</C>.
140 # The argument is a morphism $\alpha$.
141 # The output is <C>true</C> if $\alpha$ is an isomorphism,
142 # otherwise the output is <C>false</C>.
171 # The argument is a morphism $\alpha$.
172 # The output is <C>true</C> if $\alpha$ is a split monomorphism,
173 # otherwise the output is <C>false</C>.
200 # The argument is a morphism $\alpha$.
201 # The output is <C>true</C> if $\alpha$ is a split epimorphism,
202 # otherwise the output is <C>false</C>.
229 # The argument is a morphism $\alpha: a \rightarrow a$.
230 # The output is <C>true</C> if $\alpha$ is congruent to the identity of $a$,
231 # otherwise the output is <C>false</C>.
258 # The argument is a morphism $\alpha: a \rightarrow a$.
259 # The output is <C>true</C> if $\alpha^2 \sim_{a,a} \alpha$,
260 # otherwise the output is <C>false</C>.
292 #!  * <E>By integers</E>: The integer is simply a parameter that can be used to create a random mor…
293 #!  * <E>By lists</E>: The list is used when creating a random morphism would need more than one pa…
300 #! The output is a random morphism $\alpha: a \rightarrow b$ for some object
312 #! $C$ whose source is $a$.
329 #! The output is a random morphism $\alpha: a \rightarrow b$ for some object
341 #! $C$ whose source is $a$.
359 #! The output is a random morphism $\alpha: a \rightarrow b$ for some object
371 #! $C$ whose range is $b$.
388 #! The output is a random morphism $\alpha: a \rightarrow b$ for some object
400 #! $C$ whose range is $b$.
417 #! The output is a random morphism $\alpha: a \rightarrow b$ in $C$ or <C>fail</C>.
444 #! This operation is not a CAP basic operation
446 #! The output is a random morphism $\alpha: a \rightarrow b$ in $C$ or <C>fail</C>.
474 #! The output is a random morphism in $C$ or <C>fail</C>.
506 #! The output is a random morphism in $C$ or <C>fail</C>.
562 #! The argument is a morphism $\alpha: a \rightarrow b$.
563 #! The output is <C>true</C> if $\alpha = \mathrm{id}_a$,
564 #! otherwise the output is <C>false</C>.
590 #! The argument is a morphism $\alpha: a \rightarrow b$.
591 #! The output is <C>true</C> if $\alpha = 0$,
592 #! otherwise the output is <C>false</C>.
619 ## This is not a categorical property because non-endomorphisms
622 # The argument is a morphism $\alpha$.
623 # The output is <C>true</C> if $\alpha$ is an endomorphism,
624 # otherwise the output is <C>false</C>.
650 ## This is not a categorical property because non-endomorphisms
653 # The argument is a morphism $\alpha$.
654 # The output is <C>true</C> if $\alpha$ is an automorphism,
655 # otherwise the output is <C>false</C>.
705 #!  The argument <A>filter</A> is used to create a morphism type for the
706 #!  category <A>category</A>, which is then used in <C>ObjectifyMorphismForCAPWithAttributes</C>
715 #!  is created by passing a representation to <C>AddMorphismRepresentation</C>.
732 #! The output is <C>true</C> if $\alpha \sim_{a,b} \beta$,
733 #! otherwise the output is <C>false</C>.
761 #! The output is <C>true</C> if $\alpha = \beta$,
762 #! otherwise the output is <C>false</C>.
790 #! The output is <C>true</C> if $\alpha = \beta$,
791 #! otherwise the output is <C>false</C>.
828 #! The argument is a morphism $\alpha: a \rightarrow b$.
829 #! The output is <C>true</C> if $\alpha \sim_{a,b} 0$,
830 #! otherwise the output is <C>false</C>.
865 #! The output is the addition $\alpha + \beta$.
892 #! The output is the addition $\alpha - \beta$.
918 #! The argument is a morphism $\alpha: a \rightarrow b$.
919 #! The output is its additive inverse $-\alpha$.
947 #! The output is the multiplication with the ring element $r \cdot \alpha$.
986 #! The output is the zero morphism $0: a \rightarrow b$.
1025 # This is a synonym for <C>IsMonomorphism</C>.
1031 # This is a synonym for <C>IsEpimorphism</C>.
1037 #! The output is <C>true</C> if there exists an isomorphism $\iota: a \rightarrow b$
1039 #! otherwise the output is <C>false</C>.
1066 #! The output is <C>true</C> if there exists an isomorphism $\iota: b \rightarrow a$
1068 #! otherwise the output is <C>false</C>.
1109 #! In short: Returns <C>true</C> iff $\alpha$ is smaller than $\beta$.
1111 #! The output is <C>true</C> if there exists a morphism $\iota: a \rightarrow b$
1113 #! otherwise the output is <C>false</C>.
1154 #! In short: Returns <C>true</C> iff $\alpha$ is smaller than $\beta$.
1157 #! The output is <C>true</C> if there exists a morphism $\iota: b \rightarrow a$
1159 #! otherwise the output is <C>false</C>.
1192 #! The argument is an object $a$.
1193 #! The output is its identity morphism $\mathrm{id}_a$.
1221 #! The output is the composition $\beta \circ \alpha: a \rightarrow c$.
1228 #! This is a convenience method.
1229 #! The argument is a list of morphisms
1231 #! The output is the composition
1260 #! The output is the composition $\beta \circ \alpha: a \rightarrow c$.
1267 #! This is a convenience method.
1268 #! The argument is a list of morphisms
1270 #! The output is the composition
1304 #! The argument is a morphism $\alpha$.
1305 #! The output is <C>true</C> if $\alpha$ is well-defined,
1306 #! otherwise the output is <C>false</C>.
1380 #! such that there is a morphism $u: t \rightarrow k$ with
1382 #! The output is such a $u$.
1410 #! such that there is a morphism $u: c \rightarrow t$ with
1412 #! The output is such a $u$.
1440 #! The output is <C>true</C> if there exists
1443 #! Otherwise, the output is  <C>false</C>.
1471 #! The output is <C>true</C> if there exists
1474 #! Otherwise, the output is  <C>false</C>.
1501 #! The output is a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$
1503 #! Recall that a lift $\alpha / \beta: a \rightarrow b$ of $\alpha$ along $\beta$ is
1533 #! The output is a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$
1535 #! Recall that a colift $\alpha \backslash \beta: c \rightarrow b$ of $\beta$ along $\alpha$ is
1565 #! The output is <C>true</C> if there exists
1568 #! Otherwise, the output is <C>false</C>.
1597 #! The output is <C>true</C> if there exists
1600 #! Otherwise, the output is <C>false</C>.
1632 #! is a morphism $\alpha^{-1}: b \rightarrow a$ such that
1699 #!  with its previous input. To compare morphisms in the category, IsEqualForCacheForMorphism is
1700 #!  used. By default this is an alias for IsEqualForMorphismsOnMor, where fail is substituted by fa…
1702 #!  used instead. A function $F: a,b \mapsto bool$ is expected here. The output has to be
1703 #!  true or false. Fail is not allowed in this context.
1724 ## mor: x -> y
1727 ## TransportHom( mor, equality_source, equality_range ): x' -> y'
1740 …thrm{op}} \times C \rightarrow D$ (when $C$ and $D$ are Ab-categories, $H$ is assumed to be biline…
1746 #! The output is the value of the homomorphism structure on objects $H(a,b)$.
1773 #! The output is the value of the homomorphism structure on morphisms $H(\alpha, \beta )$.
1783 #! The output is the value of the homomorphism structure on morphisms $H(\alpha, \beta )$.
1809 #! The argument is a category $C$.
1810 #! The output is the distinguished object $1$ in $D$ of the homomorphism structure.
1836 #! The argument is a morphism  $\alpha: a \rightarrow a'$ in $C$.
1837 #! The output is the corresponding morphism
1867 #! The output is the corresponding morphism
1899 #! The first list $\alpha$ (the left coefficients) is a list of list of morphisms $\alpha_{ij}: A_i…
1901 #! The second list $\beta$ (the right coefficients) is a list of list of morphisms $\beta_{ij}: C_j…
1903 #! The third list $\gamma$ (the right side) is a list of morphisms $\gamma_i: A_i \rightarrow D_i$,
1905 #! The output is either