AT

A Haskell rewrite of Kenzo, a collection of algorithms for 'effective algebraic topology'. The algorithms and implementations in Kenzo were created by Francis Sergeraert, Julio Rubio Garcia, Xavier Dousson, Ana Romero and many collaborators.

Writing it from scratch myself is the only chance I have of understanding it!

Examples

See the examples/ folder.

> homotopyGroup 4 (Sphere 3)
Right ℤ/2

> homotopyGroup 6 (Sphere 3)
[ ... 30 seconds pass ... ]
Right ℤ/(2^2) ⊕ ℤ/3

> homology (Wbar (WbarDiscrete (Zmod 3)))
[ℤ,0,ℤ/3,0,ℤ/3,0,ℤ/(3^2),ℤ/3,ℤ/3,ℤ/3,ℤ/3 ⊕ ℤ/3,^C

Central Concepts of Kenzo

A simplicial set X is described by a type a, containing whatever data is required to specify X, and a type GeomSimplex a, whose elements correspond to non-degenerate simplices (in Kenzo called 'geometric simplices'). Like Kenzo we also allow a predicate on GeomSimplex a specifying when an element actually describes a geometric simplex and when it is 'spurious'.

An actual simplex of X is a geometric simplex together with a 'formal degeneracy operator', which is a list of degeneracy operators in a normal form. Face maps are implemented as functions from geometric simplices to (possibly degenerate) simplices, and the extension of these face maps to all simplices is forced by the simplicial identities.

A simplicial set is of finite type if there is a finite number of geometric simplices for each dimension, and there is a function giving a list of these simplices for any dimension n. It is not required that there are finitely many geometric simplices overall.

The normalised chain complex N(X) of X has each N(X)_n given by the free abelian group on the set of nondegenerate n-simplices of X, with the boundary of a simplex calculated similar to usual (the alternating sum of face maps), but ignoring any degenerate faces.

If C(X) is the ordinary chain complex of simplicial chains of X, the quotient map C(X) -> N(X) is a quasi-isomorphism, and so if X is of finite type, then the homology of X can be computed via N(X).

But many unavoidable simplicial sets (like K(ℤ,n) and loop spaces ΩX) are not of finite type, and so we need some other way to calculate their homology. This is where 'effective homology' comes in.

A reduction between chain complexes C and D is a strong deformation retract of chain complexes. The data of a reduction unwinds to a triple (f : C -> D, g : D -> C, h : C -> C) where f and g are degree 0, the homotopy operator h is degree 1, and certain equations involving these hold. A (strong chain) equivalence between two chain complexes C and D is a span of reductions l : E -> C and r : E -> D.

An effective homology structure on C is an equivalence between C and a chain complex F of finite type.

A simplicial set with effective homology is a simplicial set X equipped with an effective homology structure on N(X).

The point of Kenzo is that although constructions on simplicial sets sometimes do not preserve levelwise finiteness, they do extend to effective homology structures. And so if we begin with a finite simplicial complex and perform some constructions using it, then we can often compute the homology of the result even if the actual simplicial sets are now far too complicated to get a handle on.

Plan

Homological Algebra

• Definitions
  • Chain Complex
  • Bicomplex
    • Tot
  • Reduction
    • Perturbation
  • Strong Equivalence
    • Composition via span
• Constructions
  • Tensor (of chain complex)
    • Functoriality
  • Hom (of chain complex)
  • 'Bicone' (specialised pushout for surjections)
  • Bar
    • Commutative algebra structure
    • Functoriality
  • Cobar
    • Of 1-reduced
    • Of 0-reduced
    • Functoriality

Simplicial Sets

• Definitions
  • Simplicial Set
  • Simplicial Morphism
  • Simplicial Group
  • Kan Structure
  • SSet With Effective Homology
  • Principal Fibrations
  • Discrete Vector Fields
  • Coalgebra Structure on Chains
  • Algebra Structure on Chains of Groups
  • Kan Structure on Chains of Groups
• Finite Examples
  • Spheres
    • Treat S¹ separately (not 1-reduced)
  • Moore Spaces
  • Real Projective Spaces
  • Lens Spaces
• Eilenberg-MacLane Spaces
  • K(ℤ,1)
  • K(ℤ/2,1) (Can be made particularly efficient)
  • K(ℤ/p,1)
• Constructions
  • Products
    • Group structure
  • Total Space of Principal Fibration
  • Loop Space
    • Of 1-reduced
    • Of 0-reduced
    • Group structure
    • Canonical twisting function X -> GX
  • Classifying Space
    • For 0-reduced group
    • For non-reduced group
    • Special case for discrete groups
    • Group Structure
    • Canonical twisting function WG -> G
  • Suspension
    • 0-reduced
    • General Kan suspension
  • Pushouts (of 1-reduced sSets)
  • Other Finite Homotopy Colimits
  • 'Nerve' taking a ChainComplex back to a sAb?

Effective Homology

• Classifying Spaces
  • Direct Reduction of K(ℤ,1) to S¹
  • Of 0-reduced Abelian sGrps
  • Of General sGrps
• Products
  • Eilenberg-Zilber reduction
  • Use specialised contraction maps for efficiency
• Fibrations
  • Total Space from Base and Fibre ('Serre' problem)
    • 1-reduced Fibre
    • 0-reduced Fibre
  • Fibre from Base and Total Space ('Eilenberg-Moore' problem)
• Loop Space
  • 1-reduced
  • 0-reduced
• Colimits
  • Suspension
  • Pushouts (of 1-reduced sSets)
  • Finite Homotopy Colimits
• Discrete Vector Fields
  • Induced Reduction
  • Products
  • Fibrations
  • K(ℤ,1)
    • Naive but easy
    • Polynomial time but complicated
  • K(ℤ/n,1)
  • Classifying Spaces for 0-reduced sAb
• Homotopy Groups
  • Whitehead Tower (for 1-reduced sSet)
  • Postnikov Tower?
• Cohomology Operations
  • Over fields using "minimal models"

Misc TODOs

• Fix space leaks, jeez
• Pretty-printing for everything (unicode sub/superscripts in output?)
• Docs for everything
• Move this list to Github issues
• Consolidate some files? Eg. Sum, Shift into ChainComplex
• Use bit operations eg from bits-extra for degeneracy operators.
• Short-circuits: e.g. composing with zero/id for morphism/reduction/equivalence
• Make sure things are being aggressively inlined
• Make homology calculation do less work: should just need SNF of one matrix and the rank of another.
• Improve Smith normal form code, would be better to call out to some existing library instead of rolling our own. The options appear to be LinBox or FLINT. The former appears to support sparse matrices better
• Check homology of K(G,n) calculations against known results
• Add methods to produce representatives of homology classes
• Rewrite Bar to be a perturbed TensorCoalgebra?
• Rename basepoint to geomBasepoint say

Notes

• I have switched a little terminology: I believe Kenzo uses 'effective' for finite-type things and 'locally effective' for what I am calling effective things, but I find this a bit confusing.
• In Kenzo, every sSet is conflated with its chain complex of normalised chains, here I have kept the two separate.
• Avoid over-engineering the Haskell as much as possible.
• That being said, the use of Constrained.Category is a bit of a mess.
• There may be a way to unify some of the algorithms via bicomplexes and the 'generalised Eilenberg-Zilber theorem' relating the diagonal and total complexes. But the EZ-theorem only gives a strong deformation retract in special cases, in general it is just a chain homotopy equivalence.
• The classifying space functor Wbar factors through the 'total bisimplicial set' functor. But it would be difficult to describe the total functor on bisimplicial spaces algorithmically, because its definition involves the equaliser of certain face maps. So it only makes sense to implement bicomplexes and not bisimplicial sets.
• Auto-formatting the code: fourmolu -o -XTypeApplications -i $(find . -name '*.hs')
• Running Kenzo with SBCL:

  > rlwrap sbcl
  (require :asdf)
  (load "kenzo.asd")
  (asdf:load-system "kenzo")
  (in-package :kenzo)
  
  (finite-ss-table '(a b 1 c (b a)))
  

  etc.
• classes.lisp in Kenzo contains the meaning of some of the 4 letter abbreviations
  • ABSM = ABstract SiMplex
  • GMSM = GeoMetric SiMplex
  • CMBN = CoMBinatioN
  • CFFC = CoeFFiCient
  • GNRT = GeNeRaTor
  • CMPR = CoMPaRison
  • CMPRF = CoMPaRison Function
  • ICMBN = Internal-CoMBiNation
  • STRT = STRaTegy
  • bsgn = BaSe GeNerator
  • dffr = DiFFeRential
  • grmd = GRound MoDule
  • efhm = EFfective HoMology
  • idnm = IDentification NuMber
  • orgn = ORiGiN
  • vctr = VeCToR
  • intr-mrph = INTeRnal-MoRPHism (the class of actual functions implementing a morphism of simplicial sets or chain complexes)
  • sbtr = SuBTRact
  • crpr = CarRtesian PRoduct
  • BRGN = BaR GeNerator
  • TNPR = TeNsor PRoduct
• Unguessable CL functions
  • (ash x n) = bit shift x left by n
  • (add x y) and (sbtr x y) can sometimes actually be a use of the perturbation lemma(!!), depending on the types of the arguments.

References

Code:

• Kenzo homepage
• Kenzo documentation, likely out of date with the code in places
• 'Official' Kenzo mirror on GitHub, there are three different versions of the code here, I am not clear on what is gained/lost between them. There are online Jypter notebooks that let you play with Kenzo (I believe Kenzo-9) without having to figure out how to install and operate a Common Lisp environment
• Fork by Ana Romero + collaborators, has some added features over the mirror above, worth looking at resolutions.lisp and homotopy.lisp, but is missing all the discrete vector field code
• Modules written by Ana Romero, mostly to do with spectral sequences

Papers:

Everything even remotely relevant to effective algebraic topology that I can find (not all of which is relevant for implementation). Some of the documents have multiple versions; I have tried to link to the most recent in each case. Some material is repeated in different references.

[1] Simon Henry. 2026. Rewriting and presentations of quasicategories. https://doi.org/10.48550/arXiv.2608.02529

[2] Daniel Miguel Treviño. 2026. Spectral systems: New instances and algorithms. PhD thesis. Universidad de La Rioja. Retrieved from https://dialnet.unirioja.es/descarga/tesis/402493.pdf

[3] Aldo Gonzalez-Lorenzo, Alexandra Bac, and Yann-Situ Gazull. 2025. A constructive approach of Alexander duality. Journal of Applied and Computational Topology 9, 1 (2025), 2. https://doi.org/10.1007/s41468-024-00198-1

[4] Benno van den Berg and Freek Geerligs. 2025. Examples and cofibrant generation of effective Kan fibrations. Journal of Pure and Applied Algebra 229, 1 (2025), 107812. https://doi.org/10.1016/j.jpaa.2024.107812

[5] Mária Šimková. 2025. Rational homotopy equivalence. https://doi.org/10.48550/arXiv.2512.21182

[6] Francis Sergeraert. 2024. About the Kannan-Bachem algorithm. https://doi.org/10.48550/arXiv.2411.02422

[7] Miguel Angel Marco-Buzunariz and Ana Romero. 2024. Computing the homology of universal covers via effective homology and discrete vector fields. (2024). Retrieved from https://arxiv.org/abs/2409.06357

[8] Lukáš Vokřínek. 2024. Enriched categorical aspects of homological perturbation theory. https://doi.org/10.48550/arXiv.2412.21182

[9] Nikolai Mnëv. 2024. $K(Z,2)$ out of circular permutations. Retrieved from https://arxiv.org/abs/2406.01625

[10] Julián Cuevas-Rozo, Laureano Lambán, Ana Romero, and Humberto Sarria. 2023. A new method to $h$-regularize finite topological spaces. Discrete Mathematics 346, 12 (2023), 113636. https://doi.org/10.1016/j.disc.2023.113636

[11] Federico Cantero-Morán and Anibal M. Medina-Mardones. 2023. An effective proof of the cartan formula: Odd primes. Retrieved from https://arxiv.org/abs/2305.08973

[12] Marek Filakovský and Lukáš Vokřínek. 2023. Computing homotopy classes for diagrams. Discrete & Computational Geometry 70, 3 (2023), 866–920. https://doi.org/10.1007/s00454-023-00513-0

[13] Julián Cuevas-Rozo, Laureano Lambán, Ana Romero, and Humberto Sarria. 2023. Effective homological computations on finite topological spaces. Applicable Algebra in Engineering, Communication and Computing 34, 1 (2023), 33–56. https://doi.org/10.1007/s00200-020-00462-8

[14] Daniel Miguel, Andrea Guidolin, Ana Romero, and Julio Rubio. 2023. Effective spectral systems relating serre and eilenberg–moore spectral sequences. Journal of Symbolic Computation 114, (2023), 122–148. https://doi.org/10.1016/j.jsc.2022.04.014

[15] Daniel Miguel, Andrea Guidolin, Ana Romero, and Julio Rubio. 2022. A generalization of effective Serre spectral systems for $m$-multicomplexes. In Proceedings of the XVII EACA: Encuentros de álgebra computacional y aplicaciones, 2022. Castelló de la Plana, Spain, 125–128. Retrieved from https://drive.google.com/file/d/1lTgpyfDNWuIBY60lMz49cy3CIcTYkjWu/view

[16] Anibal M. Medina-Mardones. 2022. An axiomatic characterization of Steenrod’s cup-$i$ products. Retrieved from https://arxiv.org/abs/1810.06505

[17] Eduardo Sáenz-de-Cabezón and Francis Sergeraert. 2022. Discrete vector fields for monomial resolutions. In Applications of computer algebra – ACA 2022, 2022. Gebze-Istanbul, Turkey, 75–77. Retrieved from https://www.math.unm.edu/~aca/ACA/2022/scale.gtu.edu.tr/files/aca_book.pdf

[18] Anibal M. Medina-Mardones. 2022. New formulas for cup-$i$ products and fast computation of Steenrod squares. Retrieved from https://arxiv.org/abs/2105.08025

[19] Daniel Miguel, Andrea Guidolin, Ana Romero, and Julio Rubio. 2021. Constructing new spectral systems from simplicial fibrations. ACM Communications in Computer Algebra 55, 3 (September 2021), 87–91. https://doi.org/10.1145/3511528.3511534

[20] Ralph M. Kaufmann and Anibal M. Medina-Mardones. 2021. Cochain level May-Steenrod operations. Forum Math. 33, 6 (2021), 1507–1526. https://doi.org/10.1515/forum-2020-0296

[21] Andrea Guidolin and Ana Romero. 2021. Computing higher Leray-Serre spectral sequences of towers of fibrations. Foundations of Computational Mathematics 21, 4 (2021), 1023–1074. https://doi.org/10.1007/s10208-020-09475-8

[22] Andrea Guidolin, Jose Divasón, Ana Romero, and Francesco Vaccarino. 2021. Computing invariants for multipersistence via spectral systems and effective homology. Journal of Symbolic Computation 104, (2021), 724–753. https://doi.org/10.1016/j.jsc.2020.09.007

[23] Julián Cuevas-Rozo, Jose Divasón, Miguel Marco-Buzunáriz, and Ana Romero. 2021. Integration of the Kenzo system within SageMath for new algebraic topology computations. Mathematics 9, 7 (2021), 722. https://doi.org/10.3390/math9070722

[24] Matthias Franz. 2021. Szczarba’s twisting cochain and the Eilenberg-Zilber maps. Collectanea Mathematica 72, 3 (2021), 569–586. https://doi.org/10.1007/s13348-020-00299-x

[25] Ana Romero, Julio Rubio, Francis Sergeraert, and Markus Szymik. 2020. A new Kenzo module for computing the Eilenberg-Moore spectral sequence. ACM Communications in Computer Algebra 54, 2 (2020), 57–60. https://doi.org/10.1145/3427218.3427225

[26] Ruian Chen. 2020. $E_\infty$-Rings and Modules in Kan Spectral Sheaves. PhD thesis. University of Michigan. https://doi.org/2027.42/155195

[27] Stephan Zhechev. 2019. Algorithmic aspects of homotopy theory and embeddability. PhD thesis. Institute of Science; Technology Austria. https://doi.org/10.15479/AT:ISTA:6681

[28] Anibal M. Medina-Mardones. 2019. An effective proof of the cartan formula: The even prime. Retrieved from https://arxiv.org/abs/1907.12113

[29] Ana Romero, Julio Rubio, and Francis Sergeraert. 2019. An implementation of effective homotopy of fibrations. Journal of Symbolic Computation 94, (2019), 149–172. https://doi.org/10.1016/j.jsc.2018.08.001

[30] Ana Romero and Francis Sergeraert. 2019. The Eilenberg-Zilber theorem via discrete vector fields. Retrieved from https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/EZ-submitted.pdf

[31] Mikael Vejdemo-Johansson. 2018. Algorithms in $A^\infty$-algebras. Georgian Mathematical Journal 25, 4 (2018), 629–635. https://doi.org/10.1515/gmj-2018-0057

[32] Francis Sergeraert. 2018. The homological hexagonal lemma. Georgian Mathematical Journal 25, 4 (2018), 603–622. https://doi.org/10.1515/gmj-2018-0055

[33] Ana Romero and Francis Sergeraert. 2017. A Bousfield-Kan algorithm for computing the effective homotopy of a space. Foundations of Computational Mathematics 17, 5 (2017), 1335–1366. https://doi.org/10.1007/s10208-016-9322-z

[34] Ruian Chen, Igor Kriz, and Ales Pultr. 2017. Kan’s combinatorial spectra and their sheaves revisited. Theory and Applications of Categories 32, (2017), No. 39, 1363–1396. Retrieved from http://www.tac.mta.ca/tac/volumes/32/39/32-39abs.html

[35] Kathryn Hess. 2016. The Hochschild complex of a twisting cochain. Journal of Algebra 451, (2016), 302–356. https://doi.org/10.1016/j.jalgebra.2015.11.040

[36] Ana Romero and Francis Sergeraert. 2015. A combinatorial tool for computing the effective homotopy of iterated loop spaces. Discrete & Computational Geometry 53, 1 (2015), 1–15. https://doi.org/10.1007/s00454-014-9650-1

[37] Marek Filakovský. 2015. Algorithmic construction of the postnikov tower for diagrams of simplicial sets. PhD thesis. Masaryk University. Retrieved from http://www.math.muni.cz/~filakovsky/THESIS2.pdf

[38] Cyril Cohen and Anders Mörtberg. 2014. A coq formalization of finitely presented modules. In Interactive theorem proving, 2014. Springer International Publishing, 193–208. https://doi.org/10.1007/978-3-319-08970-6_13

[39] Martin Čadek, Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, and Uli Wagner. 2014. Computing all maps into a sphere. Journal of the ACM 61, 3 (2014), Art. 17, 44. https://doi.org/10.1145/2597629

[40] Marek Filakovský. 2014. Effective homology for homotopy colimit and cofibrant replacement. Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum 50, 5 (2014), 273–286. https://doi.org/10.5817/AM2014-5-273

[41] Martin Čadek, Marek Krčál, Jiří Matoušek, Lukáš Vokřínek, and Uli Wagner. 2014. Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM Journal on Computing 43, 5 (2014), 1728–1780. https://doi.org/10.1137/120899029

[42] Francis Sergeraert. 2013. Discrete vector fields and fundamental algebraic topology. Retrieved from https://www-fourier.ujf-grenoble.fr/~sergerar/Talks/13-04-Tokyo.pdf

[43] Marek Krčál, Jiří Matoušek, and Francis Sergeraert. 2013. Polynomial-time homology for simplicial Eilenberg-MacLane spaces. Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics 13, 6 (2013), 935–963. https://doi.org/10.1007/s10208-013-9159-7

[44] Ana Romero and Julio Rubio. 2012. Computing the homology of groups: The geometric way. Journal of Symbolic Computation 47, 7 (2012), 752–770. https://doi.org/10.1016/j.jsc.2011.12.007

[45] Julio Rubio and Francis Sergeraert. 2012. Constructive homological algebra and applications. Retrieved from https://arxiv.org/abs/1208.3816

[46] Danny Stevenson. 2012. Décalage and Kan’s simplicial loop group functor. Theory and Applications of Categories 26, (2012), No. 28, 768–787. Retrieved from http://www.tac.mta.ca/tac/volumes/26/28/26-28abs.html

[47] Ana Romero and Francis Sergeraert. 2012. Discrete vector fields and fundamental algebraic topology. Retrieved from https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Vector-Fields.pdf

[48] Marek Filakovský. 2012. Effective chain complexes for twisted products. Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum 48, 5 (2012), 313–322. https://doi.org/10.5817/AM2012-5-313

[49] Ana Romero and Francis Sergeraert. 2012. Effective homotopy of fibrations. Applicable Algebra in Engineering, Communication and Computing 23, 1-2 (2012), 85–100. https://doi.org/10.1007/s00200-012-0168-6

[50] Arnaud Spiwack. 2011. Verified Computing in Homological Algebra. PhD thesis. Ecole Polytechnique X. Retrieved from https://pastel.archives-ouvertes.fr/pastel-00605836

[51] Jónathan Heras. 2010. Pushout construction for the Kenzo systems. Retrieved from https://www.unirioja.es/cu/joheras/pushout/Doc/pushout.pdf

[52] Ainhoa Berciano Alcaraz, Julio Rubio, and Francis Sergeraert. 2010. A case study of $A_\infty$-structure. Georgian Mathematical Journal 17, 1 (2010), 57–77. https://doi.org/10.1515/gmj.2010.003

[53] Víctor Álvarez, José Andrés Armario, María Dolores Frau, and Pedro Real. 2010. Cartan’s constructions and the twisted Eilenberg-Zilber theorem. Georgian Mathematical Journal 17, 1 (2010), 13–23. https://doi.org/10.1515/gmj.2010.006

[54] Ana Romero. 2010. Computing the first stages of the Bousfield-Kan spectral sequence. Applicable Algebra in Engineering, Communication and Computing 21, 3 (2010), 227–248. https://doi.org/10.1007/s00200-010-0123-3

[55] Jónathan Heras. 2010. Effective homology of the pushout of simplicial sets. In Proceedings of the XII encuentros de álgebra computacional y aplicaciones, 2010. 152–156. Retrieved from https://arxiv.org/abs/1410.3651

[56] Jónathan Heras, Vico Pascual, Ana Romero, and Julio Rubio. 2010. Integrating multiple sources to answer questions in algebraic topology. In Proceedings of the 10th ASIC and 9th MKM international conference, and 17th calculemus conference on intelligent computer mathematics (AISC’10/MKM’10/calculemus’10), 2010. Springer-Verlag, Paris, France, 331–335. Retrieved from https://arxiv.org/abs/1005.0749

[57] Kathryn Hess and Andrew Tonks. 2010. The loop group and the cobar construction. Proceedings of the American Mathematical Society 138, 5 (2010), 1861–1876. https://doi.org/10.1090/S0002-9939-09-10238-1

[58] V. Álvarez, J. A. Armario, M. D. Frau, and P. Real. 2009. Algebra structures on the comparison of the reduced bar construction and the reduced $W$-construction. Communications in Algebra 37, 10 (2009), 3643–3665. https://doi.org/10.1080/00927870902747662

[59] Ana Romero, Graham Ellis, and Julio Rubio. 2009. Interoperating between computer algebra systems: Computing homology of groups with Kenzo and GAP. In ISSAC 2009—Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, 2009. ACM, New York, 303–310. https://doi.org/10.1145/1576702.1576744

[60] Francis Sergeraert. 2009. Triangulations of complex projective spaces. Retrieved from https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Mirian.pdf

[61] Mohamed Barakat and Daniel Robertz. 2008. homalg: A meta-package for homological algebra. J. Algebra Appl. 7, 3 (2008), 299–317. https://doi.org/10.1142/S0219498808002813

[62] Sebastian Thomas. 2008. The functors $\bar{W}$ and $\text{Diag} \circ \text{Nerve}$ are simplicially homotopy equivalent. Journal of Homotopy and Related Structures 3, 1 (2008), 359–378. Retrieved from https://arxiv.org/abs/0804.1082

[63] A. Romero, J. Rubio, and F. Sergeraert. 2006. Computing spectral sequences. Journal of Symbolic Computation 41, 10 (2006), 1059–1079. https://doi.org/10.1016/j.jsc.2006.06.002

[64] César Domínguez, Julio Rubio, and Francis Sergeraert. 2006. Modeling inheritance as coercion in the Kenzo system. Journal of Universal Computer Science 12, 12 (2006), 1701–1730. https://doi.org/10.3217/jucs-012-12-1701

[65] Ainhoa Berciano, María José Jiménez, and Pedro Real. 2006. Reducing computational costs in the basic perturbation lemma. In Computer algebra in scientific computing, Victor G. Ganzha, Ernst W. Mayr and Evgenii V. Vorozhtsov (eds.). Springer, Berlin, 33–48. https://doi.org/10.1007/11870814_3

[66] R. González-Díaz and P. Real. 2003. Computation of cohomology operations of finite simplicial complexes. In Homology Homotopy Appl. 83–93. https://doi.org/10.4310/HHA.2003.v5.n2.a4

[67] Alain Clément. 2002. Integral cohomology of finite Postnikov towers. PhD thesis. Université de Lausanne. Retrieved from https://doc.rero.ch/record/482

[68] M. J. Jiménez and P. Real. 2001. “Coalgebra” structures on 1-homological models for commutative differential graded algebras. In Computer algebra in scientific computing (Konstanz, 2001). Springer, Berlin, 347–361. https://doi.org/10.1007/978-3-642-56666-0_26

[69] Rocío González Díaz. 2000. Cohomology operations: A combinatorial approach. PhD thesis. University of Seville. Retrieved from https://personal.us.es/rogodi/research/tesing01.pdf

[70] Alvarez V., Armario J. A., Frau M. D., Gonzalez-Diaz R., Jiménez M. J., Real P., and Silva B. 2000. Computing “small” 1-homological models for commutative differential graded algebras. In Computer algebra in scientific computing (Samarkand, 2000). Springer, Berlin, 87–100. https://doi.org/10.1007/978-3-642-57201-2_9

[71] Pedro Real. 2000. Homological perturbation theory and associativity. Homology, Homotopy and Applications 2, (2000), 51–88. https://doi.org/10.4310/hha.2000.v2.n1.a5

[72] Rocío González-Díaz and Pedro Real. 1999. A combinatorial method for computing Steenrod squares. In J. Pure Appl. Algebra. 89–108. https://doi.org/10.1016/S0022-4049(99)00006-7

[73] P. R. Hurado, V. Álvarez, J. A. Armario, and R. González-Díaz. 1999. Algorithms in algebraic topology and homological algebra: The problem of the complexity. Zapiski Nauchnykh Seminarov POMI 258, (1999), 161–184, 358. https://doi.org/10.1023/A:1013544506151

[74] Xavier Dousson. 1999. Homologie effective des classifiants et calculs de groupes d’homotopie. PhD thesis. l’Université Joseph Fourier. Retrieved from https://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/Dousson-Xavier.pdf

[75] Julio Rubio Garcia, Francis Sergeraert, and Yvon Siret. 1999. Kenzo: A symbolic software for effective homology computation. Institut Fourier, Grenoble, France. Retrieved from https://github.com/miguelmarco/kenzo/tree/master/doc/doc_src

[76] Paul G. Goerss and John F. Jardine. 1999. Simplicial homotopy theory. Birkhäuser Verlag, Basel. https://doi.org/10.1007/978-3-0348-8707-6

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