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Thanks to Karol Szumiło’s answer to my MO question, I have added to Brown representability theorem a mention of the counterexamples for nonconnected pointed spaces and for unpointed spaces (plus a mention of Brown’s abstract categorical version).
Next task: fix the utterly horrific wikipedia page. (Edit: done!)
Yay to you and Karol!
Since my eyes fell on the name Vietoris while looking at the article, I’ll mention a fun fact I just learned today: Vietoris lived until the age of 110 years, 309 days! And his wife lived until the age of 100. Their combined ages is the greatest of any married couple on record.
I had heard that she died at the age of 108, but cannot verify my sources.
I have added various further references, with more pointers to relevant chapters, to Brown representability, hence also to generalized (Eilenberg-Steenrod) cohomology.
As for modern exposition, I like the account in section 12 of
I have finally added the statement of the full $(\infty,1)$-category theoretic version of Brown representability at Brown representability – here – and at homotopy category of an (infinity,1)-category – here.
I have further expanded at Brown representability theorem and have re-organized a little. In the process I created some auxiliary entries for ease of cross linking:
For completeness, I have typed out the proof, closely following Lurie’s proof in “Higher Algebra”. Towards the end there is room for streamlining a bit more, but I need to call it quits now.
I have spelled out in more detail the proof of the statement (now this proposition in the entry, which is the Lemma $(\star)$ on p. 114 of Higher Algebra) that in an $\infty$-category generated from cogroup objects $\{S_i\}$ a morphism $f$ is an equivalence precisely if each $Ho(\mathcal{C})(S_i,f)$ is an isomorphism.
This is the key fact needed in the proof of the Brown representability theorem to show that the object constructed there really does represent the given Brown functor. It is also the source of the crucial connectedness condition on the base spaces in the classical version of the theorem (since $S^0$ is not a cogroup and $\{S^n\}_{n \geq 1}$ generates not all pointed spaces, but just connected pointed spaces.)
But I am usure about one thing: I presently understand the proof if we assume from the outset that the set $\{S_i\}$ is closed under forming suspensions, i.e. if a set $\{S_i\}$ of cogroup objects generates $\mathcal{C}$, then I understand that a morphism is an equivalence if $Ho(\mathcal{C})(\Sigma^n S_i,f)$ is an iso for all $i\in I$ and for for all $n \in \mathbb{N}$.
Of course in the classical case this is satisfied, so there is no issue there. But in the general statement as in Higher Algebra, don’t we need to say it this way?
On p. 114 there it does say inside the proof that “Enlarging the collection $\{S_i\}$ if necessary, we may assume that this collection is stable under the formation of suspensions.”. What I am wondering about is whether we need not say this already in the statement of the lemma itself. Probably I am missing something simple.
The way I read it, since the statement ($\star$) is not stated as a separate lemma but rather as a “claim” within the proof of Theorem 1.4.1.2, it inherits all the assumptions that have been made within that proof so far, including the fact that the collection $\{S_i\}$ is stable under forming suspensions.
Okay, thanks.
I have made some steps in the writeup of the proof more explicit, for expository clarity.
For instance I have added diagrams that demonstrate the claimed existence of the iterative extensions using that $F$ takes homotopy pushouts to “weak pullbacks”:
$\array{ \left( \underset{{i \in I}\atop {\gamma \in K_i}}{\sqcup} S_i \right) &\overset{(\gamma)_{{i \in I}\atop \gamma \in K_i}}{\longrightarrow}& X_n &\overset{\eta_n}{\longrightarrow}& F \\ \downarrow &(po^{h})& \downarrow & \nearrow_{\mathrlap{\exists \eta_{n+1}}} \\ \ast &\longrightarrow& X_{n+1} } \;\;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\;\; \array{ && F(X_{n+1}) &\longrightarrow& \ast \\ &{}^{\mathllap{\exists \eta_{n+1}}}\nearrow& \downarrow^{\mathrlap{epi}} && \downarrow \\ \ast &\overset{\eta_n}{\longrightarrow}& ker\left((\gamma^\ast\right)_{{i \in I} \atop {\gamma \in K_i}}) &\longrightarrow& \ast \\ &{}_{\mathllap{\eta_n}}\searrow& \downarrow &(pb)& \downarrow \\ && F(X_n) &\underset{(\gamma^\ast)_{{i \in I} \atop {\gamma \in K_i}} }{\longrightarrow}& \underset{{i \in I}\atop {\gamma\in K_i}}{\prod}F(S_i) }$and then similarly for the total limiting extension:
$\array{ && \underset{n}{\sqcup} X_n \\ & \swarrow && \searrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' &\longleftarrow& \underset{n}{\sqcup} X_{2n} \\ & {}_{\mathllap{(\eta_{2n+1})_{n}}}\searrow& \downarrow^{\mathrlap{\exists \eta'}} & \swarrow_{\mathrlap{(\eta_{2n})_n}} \\ && F } \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; \array{ && F(X') \\ &{}^{\mathllap{\exists \eta'}}\nearrow& \downarrow^{\mathrlap{epi}} \\ &\ast \overset{(\eta_n)_n}{\longrightarrow}& \underset{\longleftarrow}{\lim}_n F(X_n) \\ & \swarrow && \searrow \\ \underset{n}{\prod}F(X_{2n+1}) && && \underset{n}{\prod}(X_{2n}) \\ & \searrow && \swarrow \\ && \underset{n}{\prod}F(X_n) } \,.$Finally I added a diagram making manifest why the map thus constructed is in indeed (not just surjective, which is obvious by construction) but also injective.
Notice that, as highlighted at the beginning of the proof, the above diagrams on the left are in $PSh(Ho(\mathcal{C}))$ instead of in $Ho(\mathcal{C})$, in order to allow the functor $F$ to be part of the diagram. I find that this move serves to make the argument much more transparent, as witnessed (I think) by these diagrams.
Where the page Brown representability theorem has:
and also weak pushouts (namely, homotopy pushouts)
presumably this is relying on the comment at weak limit that in some cases homotopy limits and weak limits relate.
That’s presumably worth a comment.
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