# Direct comparison zig-zag between cochain theories

In the paper

Mandell gives axioms for a cochain-level characterisation of ordinary cohomology theory, lifting Eilenberg–Steenrod's axioms. More accurately, he gives axioms for a cochain-level lift of absolute, unreduced cohomology, similar to Kelly (Proc. Cambridge Philos. Soc. 55 (1959) 10–22, doi:10.1017/S030500410003365X) or Hu (Portugal. Math. 19 (1960) 211–225 BNP).

The uniqueness theorem is as one would expect, modulo some fine details about the codomain of what the functor takes values in (dg $$k$$-modules vs $$E_\infty$$ k-algebras). However, he proves that the suitable functor $$T\colon \mathrm{Top} \to \mathrm{dgMod}_k$$ (a cochain theory) is unique up to natural quasi-isomorphism by constructing a zig-zag of natural quasi-isomorphisms from $$T$$ to ordinary cohomology valued in $$T(\ast)$$. I was wondering if there is a treatment out there that just constructs the connecting quasi-isomorphism zig-zag between any pair $$T_1$$ and $$T_2$$ directly. Such a thing may be hiding in lecture notes, or otherwise follow from a more sophisticated treatment. Alternatively, if there is a more-or-less obvious construction given $$T_1$$ and $$T_2$$, which isn't the concatenation of zig-zags from Mandell's proof, that would be great.

My motivation is that I will be lecturing on this, and my hope is to show that the class of cochain theories is connected, assuming one exists, and then exhibit one. This may well be singular cohomology, or it might be something else. In particular I want to remove from the proof of uniqueness the privileged position that any one construction has. The only caveat is that I won't be able to use any super-sophisticated machinery as this is a first course in algebraic topology. I'm happy to have an outline of how to unwind a sophisticated proof.

• It sounds like you're asking for an elementary statement and proof of the fact "homotopy colimit preserving functors out of Spaces are determined by their value at a point". Is that right? Unwinding I think this boils down to showing: (i) T_i composed with CW approximation is the same as T_i [by htpy axiom], (ii) T_i composed with CW approx is the same as something that looks like singular cochains with coefficients in T_i(), naturally [by excision, products, and dimension] (iii) now use an iso between T_1() and T_2(*) and extend. Apr 8, 2019 at 18:09
• but maybe what you wanted was details on (ii)? I can't tell Apr 8, 2019 at 18:11
• Sure, something like that. Albeit I can't use too much fancy machinery. Mandell's paper is just at the edge of what I can use, and I will already have to unwind a little bit of what he says. The fact he gets an explicit comparison is the bit that makes his approach appealing to me. Working at the presentation-independent $\infty$-category level makes this easy, but content-free for those not at that point. Apr 8, 2019 at 22:28
• $T_i$ composed with CW approx is the same as something that looks like singular cochains with coefficients in $T_i(\ast)$ - this is what I wanted to avoid: essentially privileging singular cohomology. I'd rather construct a direct comparison betweeen $T_1$ and $T_2$ (restricted to CW complexes, say). Apr 8, 2019 at 22:30
• @DavidRoberts: One explicit construction can be found in Definition 9.3 of dmitripavlov.org/concordance.pdf. The trick is to replace T by a naturally weakly equivalent C(T), where C(T)(S):=hocolim_n C(Δ^n⨯S). Then it is easy to define a natural weak equivalence C(T)(S)→Map(S,C(T)(*)). Apr 9, 2019 at 1:57