Making maps between simplicial modules -- information about the basic constructors

Basic constructors

• map(SimplicialModule,SimplicialModule,HashTable) -- make a map of simplicial modules
• map(SimplicialModule,SimplicialModule,ZZ) -- make the zero map or identity between simplicial modules
• map(SimplicialModule,SimplicialModule,SimplicialModuleMap) -- make a new map of simplicial modules, induced from an existing one
• id _ SimplicialModule -- the identity map of a simplicial module
• face/degeneracy maps of a simplicial module -- access the face and degeneracy maps of a simplicial module
• SimplicialModuleMap Array -- shift a simplicial module or simplicial module map
• isWellDefined(SimplicialModuleMap) -- whether a map of simplicial modules is well-defined

Important computations creating new simplicial module maps

• SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules

Canonical maps between simplicial modules

Some simplicial modules come with canonical maps. These are best accessed using inducedMap.

• kernel(SimplicialModuleMap) -- make the kernel of a map of simplicial modules
• cokernel(SimplicialModuleMap) -- make the cokernel of a map of simplicial modules
• image(SimplicialModuleMap) -- make the image of a map of simplicial modules
• coimage(SimplicialModuleMap) -- make the coimage of a map of simplicial modules
• inducedMap(SimplicialModule,SimplicialModule) -- make the map of simplicial modules induced at each term by the identity map

Random maps of simplicial modules

The method randomSimplicialMap(SimplicialModule,SimplicialModule) allows one to construct random simplicial module maps, random morphisms between simplicial modules, and random null homotopies between simplicial modules.

• isCommutative(SimplicialModuleMap) -- whether a simplicial module map commutes with the face/degeneracy maps
• isSimplicialMorphism(SimplicialModuleMap) -- whether a simplicial module map is a morphism of simplicial modules

Elementary operations on simplicial module maps

• arithmetic with simplicial module maps -- perform arithmetic operations on simplicial module maps
• SimplicialModuleMap + SimplicialModuleMap -- perform arithmetic operations on simplicial module maps
• SimplicialModuleMap | SimplicialModuleMap -- join or concatenate maps horizontally
• SimplicialModuleMap || SimplicialModuleMap -- join or concatenate maps vertically
• SimplicialModuleMap ++ SimplicialModuleMap -- direct sum of simplicial module maps
• SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules
• SimplicialModuleMap _ Array -- the composition with the canonical inclusion or projection map
• SimplicialModuleMap ^ Array -- the composition with the canonical inclusion or projection map
• truncate(List,SimplicialModuleMap) -- truncation of a simplicial module map at a specified degree or set of degrees
• RingMap SimplicialModuleMap -- apply a ring map to a map of simplicial modules
• RingMap ** SimplicialModuleMap -- tensor a map of simplicial modules along a ring map