SimplicialModule -- the class of all simplicial modules

Description

A simplicial R-module is a presheaf on the so-called Simplex Category, with values in the category of R-modules. Concretely, such objects can be viewed as nonnegatively graded R-modules equipped with certain face and degeneracy operators satisfying the simplicial identities. As an example, every R-module M can be converted into a simplicial R-module whose degree n piece is equal to M for all n, and with face/degeneracy operators simply given by the identity.

i1 : S = simplicialModule(ZZ^2,3,Degeneracy => true) --the integer 3 specifies a top degree, the option Degeneracy specifies whether or not to compute degeneracy maps

       2       2       2       2
o1 = ZZ  <-- ZZ  <-- ZZ  <-- ZZ <-- ...
                              
     0       1       2       3

o1 : SimplicialModule

The output string for a simplicial module is meant to indicate that this object is infinite in general, and the user can only compute a finite snapshot of the object. The face/degeneracy maps can be accessed using the keys dd and ss, respectively. In order to verify that the resulting face/degeneracy maps satisfy the simplicial identities, one can use the isSimplicialModule command.

i2 : keys S

o2 = {complexLength, module, ss, dd, ring, complex, topDegree, cache}

o2 : List

Functions and methods returning a Simplicial Module:

basis(List,SimplicialModule) (missing documentation)
basis(ZZ,SimplicialModule) (missing documentation)
coimage(SimplicialModuleMap) -- make the coimage of a map of simplicial modules
cokernel(SimplicialModuleMap) -- make the cokernel of a map of simplicial modules
SimplicialModule ++ SimplicialModule -- see directSum(SimplicialModule) -- direct sum of simplicial modules
extPower(ZZ,SimplicialModule) -- see extPower -- compute the Dold-Puppe extension of the exterior power functor to simplicial modules
forgetComplex(SimplicialModule) -- forget the underlying complex data of a simplicial module obtained as a Dold-Kan image
HH SimplicialModule (missing documentation)
image(SimplicialModuleMap) -- make the image of a map of simplicial modules
kernel(SimplicialModuleMap) -- make the kernel of a map of simplicial modules
minimalPresentation(SimplicialModule) -- minimal presentation of all terms in a simplicial module
prune(SimplicialModule) -- see minimalPresentation(SimplicialModule) -- minimal presentation of all terms in a simplicial module
ring(SimplicialModuleMap) -- see ring(SimplicialModule) -- access the ring of a simplicial module or a simplicial module map
Ring ** SimplicialModule -- see RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
SimplicialModule ** Ring -- see RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
SimplicialModule ** RingMap -- see RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
tensor(RingMap,SimplicialModule) -- see RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
tensor(SimplicialModule,RingMap) -- see RingMap ** SimplicialModule -- tensor a simplicial module along a ring map
RingMap SimplicialModule -- apply a ring map to a simplicial module
schurMap(List,SimplicialModule) -- see schurMap -- construct the Dold-Puppe extension of the Schur functor
simplicialModule(Complex,HashTable,HashTable,ZZ) -- see simplicialModule -- make a simplicial module
simplicialModule(Complex,HashTable,ZZ) -- see simplicialModule -- make a simplicial module
simplicialModule(HashTable,HashTable,HashTable,ZZ) -- see simplicialModule -- make a simplicial module
simplicialModule(HashTable,HashTable,ZZ) -- see simplicialModule -- make a simplicial module
Complex ** SimplicialModule -- see SimplicialModule ** SimplicialModule -- tensor product of simplicial modules
Module ** SimplicialModule -- see SimplicialModule ** SimplicialModule -- tensor product of simplicial modules
SimplicialModule ** Complex -- see SimplicialModule ** SimplicialModule -- tensor product of simplicial modules
SimplicialModule ** Module -- see SimplicialModule ** SimplicialModule -- tensor product of simplicial modules
SimplicialModule ** SimplicialModule -- tensor product of simplicial modules
simplicialModule(Ideal,ZZ) -- see simplicialModule(Module,ZZ) -- make a simplicial module associated to a complex concentrated in degree 0
simplicialModule(Module,ZZ) -- make a simplicial module associated to a complex concentrated in degree 0
simplicialModule(Ring,ZZ) -- see simplicialModule(Module,ZZ) -- make a simplicial module associated to a complex concentrated in degree 0
simplicialTensor(List) -- see simplicialTensor -- compute the simplicial tensor product and cache direct sum indices for easy access
simplicialTensor(ZZ,SimplicialModule) -- see simplicialTensor -- compute the simplicial tensor product and cache direct sum indices for easy access
simplicialTensor(SimplicialModule,SimplicialModule) (missing documentation)
source(SimplicialModuleMap) -- get the source of a map of simplicial modules
target(SimplicialModuleMap) -- get the target of a map of simplicial modules
simplicialModule(Complex) -- see The Dold-Kan Correspondence in Macaulay2 -- compute the image of a non-negatively graded complex under the Dold-Kan correspondence
simplicialModule(Complex,ZZ) -- see The Dold-Kan Correspondence in Macaulay2 -- compute the image of a non-negatively graded complex under the Dold-Kan correspondence
truncate(List,SimplicialModule) -- truncation of a simplicial module at a specified degree or set of degrees
truncate(ZZ,SimplicialModule) -- see truncate(List,SimplicialModule) -- truncation of a simplicial module at a specified degree or set of degrees

Methods that use a Simplicial Module:

components(SimplicialModule) -- list the components of a direct sum
directSum(SimplicialModule) -- direct sum of simplicial modules
exteriorInclusion(SimplicialModule) -- see exteriorInclusion -- computes the image of the 2nd exterior power into the tensor product
Symbol ^ SimplicialModule -- see face/degeneracy maps of a simplicial module -- access the face and degeneracy maps of a simplicial module
forgetDegeneracy(SimplicialModule) -- forget the data of degeneracy maps of a simplicial object
inducedMap(SimplicialModule,SimplicialModule) -- make the map of simplicial modules induced at each term by the identity map
isSimplicialModule(SimplicialModule) -- check if the simplicial identities hold for a simplicial object with degeneracy map keys
isWellDefined(SimplicialModule) -- whether a simplicial module is well-defined
map(SimplicialModule,SimplicialModule,HashTable) -- make a map of simplicial modules
map(SimplicialModule,SimplicialModule,SimplicialModuleMap) -- make a new map of simplicial modules, induced from an existing one
map(SimplicialModule,SimplicialModule,ZZ) -- make the zero map or identity between simplicial modules
naiveNorm(SimplicialModule) -- see naiveNorm(SimplicialModule,ZZ) -- compute the naive normalization of a simplicial object
naiveNorm(SimplicialModule,ZZ) -- compute the naive normalization of a simplicial object
net(SimplicialModule) (missing documentation)
normalize(SimplicialModule) -- see normalize(SimplicialModule,ZZ) -- normalization functor from simplicial modules to nonnegatively-graded chain complexes
normalize(SimplicialModule,ZZ) -- normalization functor from simplicial modules to nonnegatively-graded chain complexes
randomSimplicialMap(SimplicialModule,SimplicialModule) -- a random map of simplicial modules
ring(SimplicialModule) -- access the ring of a simplicial module or a simplicial module map
Matrix ** SimplicialModule -- see SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules
SimplicialModule ** Matrix -- create the tensor product of a simplicial module and a map of modules
SimplicialModule == SimplicialModule -- whether two simplicial modules are equal
SimplicialModule == ZZ -- see SimplicialModule == SimplicialModule -- whether two simplicial modules are equal
ZZ == SimplicialModule -- see SimplicialModule == SimplicialModule -- whether two simplicial modules are equal
SimplicialModule ^ Array -- see SimplicialModule _ Array -- the canonical inclusion or projection map of a direct sum
SimplicialModule _ Array -- the canonical inclusion or projection map of a direct sum
SimplicialModule _ Sequence -- see SimplicialModule _ ZZ -- access individual objects of a simplicial module
SimplicialModule _ ZZ -- access individual objects of a simplicial module
SimplicialModule Array -- shift a simplicial module or simplicial module map
SimplicialModule ** SimplicialModuleMap -- see SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules
SimplicialModuleMap ** SimplicialModule -- see SimplicialModuleMap ** SimplicialModuleMap -- the map of simplicial modules between tensor simplicial modules
symmetricQuotient(SimplicialModule) -- see symmetricQuotient -- computes the image of the surjection from the simplicial tensor product onto the second symmetric power of a simplicial module
topDegree(SimplicialModule) -- see topDegree -- top degree of a simplicial module

For the programmer

The object SimplicialModule is a type, with ancestor classes MutableHashTable < HashTable < Thing.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:161:0.

SimplicialModule -- the class of all simplicial modules

Description

See also

Functions and methods returning a Simplicial Module:

Methods that use a Simplicial Module:

For the programmer