Coherent homotopy commutative diagrams in algebraic topology ...

Figure 1:3.2 Weak homotopy typeMain idea is to “measure” topological spaces by means of “nice” spaces, i.e.polyhedra. Let’s denote by CW the category of spaces having the homotopytype of a polyhedron.Definition 1 A mappingf : X −→ Yis called a weak homotopy equivalence iffis a bijection for any P ∈ |CW|.[P, f] :[P, X] −→ [P, Y ]Theorem 2 (well known) For any space X there exists a polyhedron P and aweak homotopy equivalencef : P −→ X.For geometric topologists compact metric spaces are “good” (not “nice” likefinite polyhedra).Example 3 (Warsaw, or Polish, circle)X =(Figure 1).(tends to the left line like y =sin 1 x ).Crucial argument:Ȟ ∗ (X, G) ≈ H ∗ (S 1 ,G).R 2 − X homeo≈ R 2 − S 1 .However, weak homotopy type of X is the point.2

3.4 Strong shape theory3.4.1 LocalizationDefinition 5 A mappingf : X −→ Yis called a strong shape equivalence iffMap(f,P) :Map(Y,P) −→ Map(X, P)is a homotopy equivalence for any P ∈ |CW| where Map(X, P) is the simplicialset with mappingsX × ∆ n −→ Pas n-simplices (and with the evident face and degeneracy maps).Remark 6 f is a shape equivalence iff π 0 Map(f,P) is bijective for any P .Remark 7 It is enough to claim that π 0 Map(f,P) is bijective and that π 1 Map(f,P)is surjective.Remark 8 It is not known even for compact metric spaces whether any shapeequivalence is a strong shape equivalence.Analogously, a strong shape equivalence between pro-spaces is defined.Definition 9 (Marde˘síc ) The strong shape category SSH is a full subcategorySSH ⊆ pro − CWh(ss equivalences) −1iProposition 10 (Prasolov) For any category C,hpro − C ≈ inv − C (cofinal mappings) −1i .Definition 11 (Prasolov)ThestrongshapecategorySSH is a full subcategorySSH ⊆ pro − CWh((cofinal mappings) ∪ (level equivalences)) −1i3.4.2 B. GuntherGiven a space X, letX ↓ CW be a simplicial class, having as simplices homotopycommutative diagrams from X to polyhedra and coherent homotopiesX × I ? −→ P ?ThenX ↓ CW −→↓ CW4

is a so-called weak Kan fibration. Now, according to Gunther, SSH has topologicalspaces as objects, and homotopy classes of mappingsY ↓ CW −→ X ↓ CWover ↓ CW as morphisms.However, it is possible to “richly” decategorize this construction. TakeP =(holim (K) :K ∈ X ↓ CW) ,and one gets a strong shape equivalenceX −→ P.This helps to prove that Gunther’s construction is equivalent to the Lisica-Marde˘sić ’(below).3.4.3 Lisica-Marde˘sićGiven a space X, letX −→ Pbe a strong shape equivalence whereP ∈ |pro − CW| .Such an equivalence always exists. Now, they define morphisms between Xand Y as coherent morphisms between corresponding pro-objects P and Q.There is terribly difficult to define composition of such maps, to prove that thedefinition does not depend on the choice of P and Q etc.3.5 ShekutkovskiLet X be a space, and letNU =(NU : U is a normal covering) .NU is not a strict pro-object, but rather a coherent one. One can define amappingX −→ NUwhich is not a strict mapping of pro-spaces. A lot of work is needed to definemorphisms between coresponding “coherent” pro-spaces, and to prove allnecessary properties. Shekutkovski published a number of papers with a lotof mistakes, and therefore his definition cannot be considered complete. Evenmore complicated is to define homology groups (see below).5

4 HomologyThe most natural definition of (ordinary and extraordinary) strong homologywas proposed by Edwards-Hastings and later by Batanin: given a spectrum E,letH ∗ (X, E) =π ∗ holim i ω ((X ∪ pt) ∧ E) .Lisica and Marde˘sić proposed an ordinary strong homology using homotopylimits of chain complexes. There are a lot of problems: functoriality, variousproperties, spectral sequences etc. One needs also to prove that, in the caseE = K(G),the definition above is equivalent to Lisica-Marde˘sić definition. That was doneby Prasolov [Pra01].5 Challenges• Define (topologically or multi-categorically) a coherent pro-space.• Define coherent morphisms between coherent pro-spaces.• Define a strong shape equivalence between coherent pro-spaces.• Define an (ordinary or extraordinary) homology for coherent pro-spaces6

References[Bat93] M. A. Batanin. Coherent categories with respect to monads andcoherent prohomotopy theory. Cahiers Topologie Géom. DifférentielleCatég., 34(4):279—304, 1993.[Bat97][CP97][EH76][Gün91]Mikhail A. Batanin. Categorical strong shape theory. Cahiers TopologieGéom. Différentielle Catég., 38(1):3—66, 1997.Jean-Marc Cordier and Timothy Porter. Homotopy coherent categorytheory. Trans. Amer. Math. Soc., 349(1):1—54, 1997.David A. Edwards and Harold M. Hastings. Čech and Steenrod homotopytheories with applications to geometric topology. Springer-Verlag,Berlin, 1976. Lecture Notes in Mathematics, Vol. 542.Bernd Günther. Comparison of the coherent pro-homotopy theoriesof Edwards-Hastings, Lisica-Mardešić and Günther. Glas.Mat.Ser.III, 26(46)(1-2):141—176, 1991.[Gün92a] Bernd Günther. The use of semisimplicial complexes in strong shapetheory. Glas.Mat.Ser.III, 27(47)(1):101—144, 1992.[Gün92b] Bernd Günther. The Vietoris system in strong shape and stronghomology. Fund. Math., 141(2):147—168, 1992.[Mar00][MP98][Pra01]Sibe Mardešić. Strong shape and homology. Springer Monographs inMathematics. Springer-Verlag, Berlin, 2000.Sibe Mardešić and Andrei V. Prasolov. On strong homology of compactspaces. Topology Appl., 82(1-3):327—354, 1998. Special volumein memory of Kiiti Morita.Andrei V. Prasolov. Extraordinary strong homology. Topology Appl.,113(1-3):249—291, 2001. Geometric topology: Dubrovnik 1998.[Pra05] Andrei V. Prasolov. Non-additivity of strong homology. TopologyAppl., 153(2-3):493—527, 2005.[She01] Nikita Shekutkovski. From inverse to coherent systems. TopologyAppl., 113(1-3):293—307, 2001. Geometric topology: Dubrovnik 1998.[She04] Nikita Shekutkovski. Shift and coherent shift in inverse systems.Topology Appl., 140(1):111—130, 2004.[Str05] L. Stramaccia. Groupoids and strong shape. Topology Appl., 153(2-3):528—539, 2005.7