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Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and attaching each $X_i\times \{1\}$ to $X_{i+1}\times \{0\}$ via the map $X_i\to X_{i+1}$.

What is an example (if there is one) in which the homotopy direct limit of such a sequence is not homotopy equivalent to the regular direct limit (in the category of topological spaces) of that sequence?

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  • $\begingroup$ Could you give a citation for each usage? In particular, what do you mean when you say "direct limit"? (i.e. what category are you talking about) $\endgroup$ – Dorebell Aug 19 '14 at 4:00
  • $\begingroup$ This is in the category of topological spaces, probably the algebraic topologists' category to be precise, so something like compactly generated spaces with nondegenerate basepoint. $\endgroup$ – Kevin Carlson Aug 19 '14 at 4:02
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If your maps are cofibrations, then the direct limit and the homotopy direct limit coincide (up to homotopy equivalence). So to find a counterexample, we must look at some less nice sequences of maps.

For instance, consider the map $f : S^1 \to S^1$ defined by $e^{\theta i} \mapsto e^{2 \theta i}$. This is not a cofibration, obviously. The direct limit of the sequence $$S^1 \stackrel{f}{\to} S^1 \stackrel{f}{\to} S^1 \stackrel{f}{\to} S^1 \to \cdots$$ is the quotient of $S^1$ as a topological group by the subgroup $G = \{ e^{2^{-n} k \pi i} : n \in \mathbb{N}, k \in \mathbb{N} \}$. This rather monstrous-seeming space is actually the indiscrete topological space on $\left| S^1 \right|$-many points, hence is contractible.

On the other hand, the homotopy direct limit of the sequence is not contractible. Indeed, $\pi_1$ sends homotopy direct limits to direct limits, and the corresponding sequence of $\pi_1$ is $$\mathbb{Z} \stackrel{2 \times}{\to} \mathbb{Z} \stackrel{2 \times}{\to} \mathbb{Z} \stackrel{2 \times}{\to} \mathbb{Z} \to \cdots$$ so $\pi_1$ of the direct limit is additive group of the ring $\mathbb{Z} [\frac{1}{2}]$.

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Let $X_n=\{0\}\cup[\frac1n,1]$ and $X_n\to X_{n+1}$ the obvious injection. Then $\mathop{\mathrm{dirlim}} X_n=[0,1]$ whereas the homotopy direct limit of $X_n$ is disconnected.

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    $\begingroup$ Are you sure that the direct limit in the category of topological spaces is [0,1]? Set-theoretically it's [0,1], but I believe the topology on it makes {0} an open set, because it's an open in each of the Xn. $\endgroup$ – user64480 Aug 19 '14 at 18:48
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    $\begingroup$ @user64480 is correct. At least, $[0,1]$ is not the direct limit, which can be seen with an argument involving universal properties. $\endgroup$ – Simon Rose Aug 19 '14 at 19:05

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