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AT

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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
    • Pushouts (of 1-reduced sSets)
    • Other Finite Homotopy Colimits
    • 'Nerve' taking a ChainComplex back to a sAb?

Effective Homology

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:

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] About the Kannan-Bachem algorithm: 2024. https://arxiv.org/abs/2411.02422.

[2] Álvarez, V. et al. 2009. Algebra structures on the comparison of the reduced bar construction and the reduced $W$-construction. Communications in Algebra. 37, 10 (2009), 3643–3665. DOI:https://doi.org/[10.1080/00927870902747662](https://doi.org/10.1080/00927870902747662).

[3] Álvarez, V. et al. 2010. Cartan’s constructions and the twisted Eilenberg-Zilber theorem. Georgian Mathematical Journal. 17, 1 (2010), 13–23. DOI:https://doi.org/[10.1515/gmj.2010.006](https://doi.org/10.1515/gmj.2010.006).

[4] An axiomatic characterization of Steenrod’s cup-$i$ products: 2022. https://arxiv.org/abs/1810.06505.

[5] An effective proof of the cartan formula: Odd primes: 2023. https://arxiv.org/abs/2305.08973.

[6] An effective proof of the cartan formula: The even prime: 2019. https://arxiv.org/abs/1907.12113.

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

[8] Berciano, A. et al. 2006. Reducing computational costs in the basic perturbation lemma. Computer algebra in scientific computing. V.G. Ganzha et al., eds. Springer, Berlin. 33–48.

[9] Berciano Alcaraz, A. et al. 2010. A case study of $A_\infty$-structure. Georgian Mathematical Journal. 17, 1 (2010), 57–77. DOI:https://doi.org/[10.1515/gmj.2010.003](https://doi.org/10.1515/gmj.2010.003).

[10] Čadek, M. et al. 2014. Computing all maps into a sphere. Journal of the ACM. 61, 3 (2014), Art. 17, 44. DOI:https://doi.org/[10.1145/2597629](https://doi.org/10.1145/2597629).

[11] Čadek, M. et al. 2014. Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM Journal on Computing. 43, 5 (2014), 1728–1780. DOI:https://doi.org/[10.1137/120899029](https://doi.org/10.1137/120899029).

[12] Chen, R. 2020. $E_\infty$-Rings and Modules in Kan Spectral Sheaves. University of Michigan.

[13] Chen, R. et al. 2017. Kan’s combinatorial spectra and their sheaves revisited. Theory and Applications of Categories. 32, (2017), No. 39, 1363–1396.

[14] Clément, A. 2002. Integral cohomology of finite Postnikov towers. Université de Lausanne.

[15] Cohen, C. and Mörtberg, A. 2014. A coq formalization of finitely presented modules. Interactive theorem proving (2014), 193–208.

[16] Constructive homological algebra and applications: 2012. https://arxiv.org/abs/1208.3816.

[17] Cuevas-Rozo, J. et al. 2023. Effective homological computations on finite topological spaces. Applicable Algebra in Engineering, Communication and Computing. 34, 1 (2023), 33–56. DOI:https://doi.org/[10.1007/s00200-020-00462-8](https://doi.org/10.1007/s00200-020-00462-8).

[18] Díaz, R.G. 2000. Cohomology operations: A combinatorial approach. University of Seville.

[19] Discrete vector fields and fundamental algebraic topology: 2012. https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Vector-Fields.pdf.

[20] Domínguez, C. et al. 2006. Modeling inheritance as coercion in the Kenzo system. Journal of Universal Computer Science. 12, 12 (2006), 1701–1730. DOI:https://doi.org/[10.3217/jucs-012-12-1701](https://doi.org/10.3217/jucs-012-12-1701).

[21] Dousson, X. 1999. Homologie effective des classifiants et calculs de groupes d’homotopie. l’Université Joseph Fourier.

[22] Eilenberg, S. and Mac Lane, S. 1953. On the groups $H(\Pi,n)$. I. Ann. of Math. (2). 58, (1953), 55–106. DOI:https://doi.org/[10.2307/1969820](https://doi.org/10.2307/1969820).

[23] Eilenberg, S. and Mac Lane, S. 1954. On the groups $H(\Pi,n)$. II. Methods of computation. Ann. of Math. (2). 60, (1954), 49–139. DOI:https://doi.org/[10.2307/1969702](https://doi.org/10.2307/1969702).

[24] Filakovský, M. 2015. Algorithmic construction of the postnikov tower for diagrams of simplicial sets. Masaryk University.

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

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

[27] Forman, R. 1998. Morse theory for cell complexes. Advances in Mathematics. 134, 1 (1998), 90–145. DOI:https://doi.org/[10.1006/aima.1997.1650](https://doi.org/10.1006/aima.1997.1650).

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

[29] Goerss, P.G. and Jardine, J.F. 1999. Simplicial homotopy theory. Birkhäuser Verlag, Basel.

[30] González-Díaz, R. and Real, P. 1999. A combinatorial method for computing Steenrod squares. J. Pure Appl. Algebra. 89–108.

[31] González-Díaz, R. and Real, P. 2003. Computation of cohomology operations of finite simplicial complexes. Homology Homotopy Appl. 83–93.

[32] Gonzalez-Lorenzo, A. et al. 2025. A constructive approach of Alexander duality. Journal of Applied and Computational Topology. 9, 1 (2025), 2. DOI:https://doi.org/[10.1007/s41468-024-00198-1](https://doi.org/10.1007/s41468-024-00198-1).

[33] Gugenheim, V.K.A.M. et al. 1991. Perturbation theory in differential homological algebra. II. Illinois J. Math. 35, 3 (1991), 357–373. DOI:https://doi.org/[10.1215/ijm/1255987784](https://doi.org/10.1215/ijm/1255987784).

[34] Gugenheim, V.K.A.M. and Lambe, L.A. 1989. Perturbation theory in differential homological algebra. I. Illinois J. Math. 33, 4 (1989), 566–582. DOI:https://doi.org/[10.1215/ijm/1255988571](https://doi.org/10.1215/ijm/1255988571).

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

[36] Heras, J. 2010. Effective homology of the pushout of simplicial sets. Proceedings of the XII encuentros de álgebra computacional y aplicaciones (2010), 152–156.

[37] Heras, J. et al. 2010. Integrating multiple sources to answer questions in algebraic topology. Proceedings of the 10th ASIC and 9th MKM international conference, and 17th calculemus conference on intelligent computer mathematics (Paris, France, 2010), 331–335.

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

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

[40] Huebschmann, J. and Kadeishvili, T. 1991. Small models for chain algebras. Math. Z. 207, 2 (1991), 245–280. DOI:https://doi.org/[10.1007/BF02571387](https://doi.org/10.1007/BF02571387).

[41] Hurado, P.R. et al. 1999. Algorithms in algebraic topology and homological algebra: The problem of the complexity. Zapiski Nauchnykh Seminarov POMI. 258, (1999), 161–184, 358. DOI:https://doi.org/[10.1023/A:1013544506151](https://doi.org/10.1023/A:1013544506151).

[42] Jiménez, M.J. and Real, P. 2001. “Coalgebra” structures on 1-homological models for commutative differential graded algebras. Computer algebra in scientific computing (Konstanz, 2001). Springer, Berlin. 347–361.

[43] Kadeishvili, T. and Saneblidze, S. 1998. Iterating the bar construction. Georgian Mathematical Journal. 5, 5 (1998), 441–452. DOI:https://doi.org/[10.1023/B:GEOR.0000008115.37751.62](https://doi.org/10.1023/B:GEOR.0000008115.37751.62).

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

[45] Krčál, M. et al. 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. DOI:https://doi.org/[10.1007/s10208-013-9159-7](https://doi.org/10.1007/s10208-013-9159-7).

[46] Lambe, L. and Stasheff, J. 1987. Applications of perturbation theory to iterated fibrations. Manuscripta Mathematica. 58, 3 (1987), 363–376. DOI:https://doi.org/[10.1007/BF01165893](https://doi.org/10.1007/BF01165893).

[47] Marco-Buzunariz, M.A. and Romero, A. 2024. Computing the homology of universal covers via effective homology and discrete vector fields. (2024).

[48] Miguel, D. et al. 2022. A generalization of effective Serre spectral systems for $m$-multicomplexes. Proceedings of the XVII EACA: Encuentros de álgebra computacional y aplicaciones (Castelló de la Plana, Spain, 2022), 125–128.

[49] Miguel, D. et al. 2021. Constructing new spectral systems from simplicial fibrations. ACM Communications in Computer Algebra. 55, 3 (Sept. 2021), 87–91. DOI:https://doi.org/[10.1145/3511528.3511534](https://doi.org/10.1145/3511528.3511534).

[50] Miguel, D. et al. 2023. Effective spectral systems relating serre and eilenberg–moore spectral sequences. Journal of Symbolic Computation. 114, (2023), 122–148. DOI:https://doi.org/[10.1016/j.jsc.2022.04.014](https://doi.org/10.1016/j.jsc.2022.04.014).

[51] Mnëv, N. 2024. $K(Z,2)$ out of circular permutations.

[52] Morace, F. 1994. Cochaînes de brown et transformation d’Eilenberg-Mac Lane: Réécriture en dimension deux et homologie. Paris 7.

[53] Morace, F. and Prouté, A. 1994. Brown’s natural twisting cochain and the Eilenberg-Mac Lane transformation. J. Pure Appl. Algebra. 97, 1 (1994), 81–89. DOI:https://doi.org/[10.1016/0022-4049(94)90040-X](https://doi.org/10.1016/0022-4049(94)90040-X).

[54] New formulas for cup-$i$ products and fast computation of Steenrod squares: 2022. https://arxiv.org/abs/2105.08025.

[55] Pushout construction for the Kenzo systems: 2010. https://www.unirioja.es/cu/joheras/pushout/Doc/pushout.pdf.

[56] Real Jurado, P. 1993. Algoritmos de cálculo de homología efectiva de los espacios clasificantes. Universidad de Sevilla, Departamento de Geometría y Topología.

[57] Real, P. 1996. An algorithm computing homotopy groups. Math. Comput. Simulation. 42, 4-6 (1996), 461–465. DOI:https://doi.org/[10.1016/S0378-4754(96)00021-3](https://doi.org/10.1016/S0378-4754(96)00021-3).

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

[59] Real, P. 1996. On the computability of the Steenrod squares. Ann. Univ. Ferrara Sez. VII (N.S.). 42, (1996), 57–63 (1998). DOI:https://doi.org/[10.1007/BF02955020](https://doi.org/10.1007/BF02955020).

[60] Romero, A. et al. 2020. A new Kenzo module for computing the Eilenberg-Moore spectral sequence. ACM Communications in Computer Algebra. 54, 2 (2020), 57–60. DOI:https://doi.org/[10.1145/3427218.3427225](https://doi.org/10.1145/3427218.3427225).

[61] Romero, A. et al. 2019. An implementation of effective homotopy of fibrations. Journal of Symbolic Computation. 94, (2019), 149–172. DOI:https://doi.org/[10.1016/j.jsc.2018.08.001](https://doi.org/10.1016/j.jsc.2018.08.001).

[62] Romero, A. et al. 2006. Computing spectral sequences. Journal of Symbolic Computation. 41, 10 (2006), 1059–1079. DOI:https://doi.org/[10.1016/j.jsc.2006.06.002](https://doi.org/10.1016/j.jsc.2006.06.002).

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

[64] Romero, A. et al. 2009. Interoperating between computer algebra systems: Computing homology of groups with Kenzo and GAP. ISSAC 2009—Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation (2009), 303–310.

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

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

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

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

[69] Rubio Garcia, J. et al. 1999. Kenzo: A symbolic software for effective homology computation. Institut Fourier.

[70] Rubio, J. and Sergeraert, F. 1993. Locally effective objects and algebraic topology. Computational algebraic geometry (Nice, 1992). F. Eyssette and A. Galligo, eds. Birkhäuser Boston, Boston, MA. 235–251.

[71] Sáenz-de-Cabezón, E. and Sergeraert, F. 2022. Discrete vector fields for monomial resolutions. Applications of computer algebra – ACA 2022 (Gebze-Istanbul, Turkey, 2022), 75–77.

[72] Sergeraert, F. 2013. Discrete vector fields and fundamental algebraic topology.

[73] Sergeraert, F. 1987. Homologie effective. I. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics. 304, 11 (1987), 279–282.

[74] Sergeraert, F. 1987. Homologie effective. II. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics. 304, 12 (1987), 319–321.

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

[76] Spiwack, A. 2011. Verified Computing in Homological Algebra. Ecole Polytechnique X.

[77] Stevenson, D. 2012. Décalage and Kan’s simplicial loop group functor. Theory and Applications of Categories. 26, (2012), No. 28, 768–787.

[78] The Eilenberg-Zilber theorem via discrete vector fields: 2019. https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/EZ-submitted.pdf.

[79] Thomas, S. 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.

[80] Triangulations of complex projective spaces: 2009. https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Mirian.pdf.

[81] V., A. et al. 2000. Computing “small” 1-homological models for commutative differential graded algebras. Computer algebra in scientific computing (Samarkand, 2000). Springer, Berlin. 87–100.

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

[83] Zhechev, S. 2019. Algorithmic aspects of homotopy theory and embeddability. Institute of Science; Technology Austria.

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Effective Algebraic Topology in Haskell

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