Overview

This project consisted in improving the interface between the Kenzo Program and SageMath, whose initial version was worked in ticket # 27880 (that ticket makes it possible to communicate both computer algebra programs and enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of infinite nature). Specifically, our purpose was to combine the functions and techniques of algebraic topology for simplicial sets developed in Kenzo and those existing in the module SimplicialSets of SageMath, in order to complement and enrich the functionalities for computing invariants of such spaces in SageMath.

Trac Ticket

Bearing in mind the SageMath development process by mean of an issue tracking system called Trac, we have created a ‘‘ticket’’. This is the name given for an item on the server, where anyone can post on the trac server if a bug is found in SageMath, if new code is ready to be submitted or you want to review new code already written but not yet included in SageMath or if there are corrections for the documentation. Anyone may search or browse the tickets and for a list of recent changes, Sage trac timeline must be visited.

#29879 Linking Kenzo simplicial sets to SageMath ones

This is the list of commits associated to the work done in the project:

Commit message	Author	Age	Files	Lines
Descriptions deleted in lines # optional - kenzo	gh-jcuevas-rozo	2020-08-23	4	-23/+23
doctests depending on Kenzo marked as optional	gh-jcuevas-rozo	2020-08-11	7	-111/+156
Documentation added about ‘_kenzo_repr’ attribute	gh-jcuevas-rozo	2020-08-03	6	-40/+795
classes KenzoAbstractSimplex and KenzoCRPRSimplex added	gh-jcuevas-rozo	2020-07-28	3	-222/+339
_kenzo_repr attribute added to ProductOfSimplicialSets, WedgeOfSimplicialSets and SmashProductOfSimplicialSets	gh-jcuevas-rozo	2020-07-22	1	-0/+29
Updated	gh-jcuevas-rozo	2020-07-09	2	-4/+1
_kenzo_repr added to operations of ChainComplexMorphisms	gh-jcuevas-rozo	2020-07-09	1	-18/+12
class KenzoCombination created; classes KenzoChainComplex and KenzoChainComplexMorphism callable	gh-jcuevas-rozo	2020-07-09	1	-23/+528
_kenzo_repr for compositions of ChainComplexMorphisms added	gh-jcuevas-rozo	2020-07-03	2	-64/+158
_kenzo_repr attribute added to ChainComplex and ChainComplexMorphism	gh-jcuevas-rozo	2020-07-02	3	-7/+32
Modifying _kenzo_repr attribute (not optional)	gh-jcuevas-rozo	2020-06-26	3	-50/+34
Parameter kenzo_repr added to special simplicial sets	gh-jcuevas-rozo	2020-06-24	4	-17/+46
KAbstractSimplex function changed	gh-jcuevas-rozo	2020-06-21	1	-1/+2
Optional parameter kenzo_repr added to AbstractSimplex_class.init and SimplicialSet_finite.init	gh-jcuevas-rozo	2020-06-18	2	-16/+33
Minor improvements to kenzo.py	jcuevas-rozo	2020-06-16	1	-5/+3

Modified files

In the files we were modifying along the development of the project, the next three lines were added to the respective preambles:

from sage.interfaces import kenzo
from sage.features.kenzo import Kenzo
kenzo_is_present = Kenzo().is_present()

Above lines allow us to know if the Kenzo system is installed and they provide a condition to the Kenzo representations of the Sage objects. The following items show some descriptions of the added functions and changes made on the Kenzo files and SageMath files:

kenzo.py

This is the file where the wrapping functions are defined and implemented for Kenzo objects.

• The building blocks for simplicial sets are abstract simplices. These objects are implemented in Kenzo in a special type ABSM. We considered important to create a class KenzoAbstractSimplex as a wrapper for this type of Kenzo objects. Different methods were implemented for this class: the non-degenerate part, the list of degeneracy operators applied to it (if they exist) and the cartesian product with other abstract simplices.

• In Kenzo, the non-degenerate simplices of a cartesian product of simplicial sets are represented internally by an object of type CRPR. In order to wrap this objects, the class KenzoCRPRSimplex was created and a method to return the factors that generate the abstract simplex in the cartesian product was implemented.

• Combinations in Kenzo are objects of type CMBN representing a sum of terms in a chain complex. The class KenzoCombination was implemented to wrap these objects and methods wrapping Kenzo functions dealing with combinations were also created: the opposite of a combination, the sum, substraction and scalar multiplication. The function Kenzocmbn was implemented to create an instance of KenzoCombination from an ECL object representing a combination.

• In the class KenzoChainComplexMorphism we have added methods for the opposite of a morphism, the sum, substraction, composition and scalar multiplication of morphisms.

• For simplicial set morphisms, we have created the class KenzoSimplicialSetMorphism to wrap simplicial set morphisms in Kenzo. Methods for computing the cone, the suspension and the pushout of simplicial set morphisms have been created to deal with these objects. The function KSimplicialSetMorphism was implemented to create a KenzoSimplicialSetMorphism from a SimplicialSetMorphism in SageMath.

• The functions KenzoId and IdNumber were implemented in order to use the slot :idnm in some Kenzo objects (like, simplicial sets, chain complexes, morphisms, spectral sequences,…) to identify and recover the respective object in Kenzo. Also, the method orgn was implemented in classes were the Kenzo objects have :orgn slot.

sage-interface.lisp

This is a file where the Kenzo functions are implemented in order to be imported in the file kenzo.py.

• The function KDFFR was added to provide the slot :intr-dffr of the KenzoSimplicialSet in SageMath obtained by applying the function KChainComplex (in kenzo.py) to a ChainComplex in Sage.

• The function KINTR was added to provide the slot :intr of the KenzoChainComplexMorphism in SageMath obtained by applying the function KMorphismChainComplex (in kenzo.py) to a ChainComplexMorphism in Sage. In a similar way, the function KSINTR was added to provide the slot :sintr of the KenzoSimplicialMorphism in SageMath obtained by applying the function KSimplicialSetMorphism (in kenzo.py) to a SimplicialSetMorphism in Sage.

• The function KCHAINCOMPLEXMORPHISM-AUX was added to construct a chain complex morphism in Kenzo from the information given by the dictionary of matrices defining a ChainComplexMorphism in Sage. In a similar way, the function KSIMPLICIALSETMORPHISM-AUX was added to construct a simplicial set morphism in Kenzo from the information given by the dictionary of matrices defining a SimplicialSetMorphism in Sage

• Some wrapper functions were implemented in order to use combinations (type CMBN in Kenzo) in SageMath (CMBN-AUX, DFFR-AUX1, EVALUATE-CMBN) and wrapper functions to deal with abstract simplices (type ABSM in Kenzo) and abstract simplices of cartesian products (type CRPR in Kenzo) in SageMath were added (ABSM-AUX, DEGENERATE-P, NON-DEGENERATE-P, CRPR-ABSM-AUX, ABSM1, ABSM2, KABSTRACTSIMPLEX-AUX) among other functions.

chain_complex.py

• The _kenzo_repr_ attribute was added to the class ChainComplex_class: when a ChainComplex in Sage has grading group the set of integer numbers and the degree of differential is -1, its Kenzo representation is constructed (these are restrictions given by the Kenzo system).

chain_complex_morphism.py

• The _kenzo_repr_ attribute was added to the class ChainComplexMorphism: if the source complex and the target complex have _kenzo_repr_ attributes, the Kenzo representation of the morphism is constructed.

• Different operations between ChainComplexMorphisms have Kenzo representations: the opposite of a morphism, sum of morphisms, composition of morphisms, multiplication by an integer number, substraction of morphisms. All of these operations have _kenzo_repr_ attributes whenever the involved morphisms have Kenzo representations.

simplicial_set.py

• The _kenzo_repr_ attribute was added to the class AbstractSimplex_class. This needed the implementation of the class KenzoAbstractSimplex in kenzo.py. Different methods of this class have the respective Kenzo representations: degeneracies, nondegenerate part of an abstract simplex, the product of abstract simplices (this needed the implementation of the class KenzoCRPRSimplex in kenzo.py).

• In the class SimplicialSet_arbitrary, the join method is not implemented, but in Kenzo this operation is available. The join of simplicial sets is created as a KenzoSimplicialSet whenever the onvolved simplicial sets have _kenzo_repr_ attributes. In the class SimplicialSet_finite, the _kenzo_repr_ attribute was also added.

simplicial_set_constructions.py

• In class PullbackOfSimplicialSets_finite, the stored information in attribute self._translation is used for “translate” the involved simplices to their Kenzo equivalent abstract simplices. Since the product of simplicial sets is implemented in SageMath as a pullback, this translation allows us to associate the _kenzo_repr_ attribute to the class ProductOfSimplicialSets.

• In classes PushoutOfSimplicialSets_finite and PushoutOfSimplicialSets, the _kenzo_repr_ attribute was added bearing in mind that (by now) only pushouts of two morphisms are constructed in Kenzo.

• The _kenzo_repr_ attribute was also added to the classes SmashProductOfSimplicialSets_finite and SmashProductOfSimplicialSets, where the Kenzo analog is constructed if all the involved factors have their Kenzo representation. The same was made for classes WedgeOfSimplicialSets and WedgeOfSimplicialSets_finite.

• If a simplicial set has _kenzo_repr_ attribute, its suspension have a Kenzo representation (implemented in classes SuspensionOfSimplicialSet and SuspensionOfSimplicialSet_finite).

simplicial_set_examples.py

• In the function Sphere was assign the _kenzo_repr_ attribute in order to construct a Kenzo sphere, which is possible when the dimension of the sphere is less than 15 (the maximal dimension allowed in Kenzo system). Also, in Simplex function the Kenzo analog (DELTA function) was added as the _kenzo_repr_ attribute. In the function RealProjectiveSpace the _kenzo_repr_ attribute was assigned allowing the construction of the analog finite or infinite dimensional real projective space in Kenzo.

simplicial_set_morphism.py

• In the class SimplicialSetMorphism, the _kenzo_repr_ attribute is added when the domain and the codomain simplicial sets have Kenzo representations. Since the definition of a simplicial set morphism in SageMath is storaged in self._dictionary, we have used this information to construct the Kenzo representation of this morphisms

• Some constructions involving simplicial set morphisms, like cones, suspensions and pushouts, which are implemented in SageMath, were assigned their respective _kenzo_repr_ attributes.