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Qiaochu Yuan
Qiaochu Yuan
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This is not really an answer. The inclusion of homotopy $1$-types into homotopy types has a left adjoint $\tau_{\le 1}$ given by truncation, and hence preserves homotopy limits (e.g. products). The question at hand is whether there is a homotopy-theoretic way to see that it also preserves certain homotopy colimits, namely a certain shape of homotopy pushout.

One might ask more generally what happens when $1$ is replaced by $n$, and here I believe it's not true that the inclusion of homotopy $n$-types preserves homotopy pushouts; e.g. I don't believe it's true that the wedge sum of two homotopy $n$-types is another homotopy $n$-type. (The first counterexample that comes to mind only works rationally: rationally $S^2$ is a $2$-type, but $S^2 \vee S^2$ has rational homotopy in arbitrarily high degrees.)

So there's something special about the inclusion of homotopy $1$-types. At this point you might be happy to learn that it's possible to develop covering space theory for groupoids without any reference to topological spaces. I think this is already enough to prove the desired result, by a construction, at the level of groupoids, of the universal cover of $BG \vee BH$. In fact it ought to be possible to completely describe the covering theory of $BG \vee BH$ at the level of groupoids, leading to the Kurosh subgroup theorem. For some indication of how this might be done see this blog postthis blog post, which has some pictures of the special case $G = C_2, H = C_3$.

This is not really an answer. The inclusion of homotopy $1$-types into homotopy types has a left adjoint $\tau_{\le 1}$ given by truncation, and hence preserves homotopy limits (e.g. products). The question at hand is whether there is a homotopy-theoretic way to see that it also preserves certain homotopy colimits, namely a certain shape of homotopy pushout.

One might ask more generally what happens when $1$ is replaced by $n$, and here I believe it's not true that the inclusion of homotopy $n$-types preserves homotopy pushouts; e.g. I don't believe it's true that the wedge sum of two homotopy $n$-types is another homotopy $n$-type. (The first counterexample that comes to mind only works rationally: rationally $S^2$ is a $2$-type, but $S^2 \vee S^2$ has rational homotopy in arbitrarily high degrees.)

So there's something special about the inclusion of homotopy $1$-types. At this point you might be happy to learn that it's possible to develop covering space theory for groupoids without any reference to topological spaces. I think this is already enough to prove the desired result, by a construction, at the level of groupoids, of the universal cover of $BG \vee BH$. In fact it ought to be possible to completely describe the covering theory of $BG \vee BH$ at the level of groupoids, leading to the Kurosh subgroup theorem. For some indication of how this might be done see this blog post, which has some pictures of the special case $G = C_2, H = C_3$.

This is not really an answer. The inclusion of homotopy $1$-types into homotopy types has a left adjoint $\tau_{\le 1}$ given by truncation, and hence preserves homotopy limits (e.g. products). The question at hand is whether there is a homotopy-theoretic way to see that it also preserves certain homotopy colimits, namely a certain shape of homotopy pushout.

One might ask more generally what happens when $1$ is replaced by $n$, and here I believe it's not true that the inclusion of homotopy $n$-types preserves homotopy pushouts; e.g. I don't believe it's true that the wedge sum of two homotopy $n$-types is another homotopy $n$-type. (The first counterexample that comes to mind only works rationally: rationally $S^2$ is a $2$-type, but $S^2 \vee S^2$ has rational homotopy in arbitrarily high degrees.)

So there's something special about the inclusion of homotopy $1$-types. At this point you might be happy to learn that it's possible to develop covering space theory for groupoids without any reference to topological spaces. I think this is already enough to prove the desired result, by a construction, at the level of groupoids, of the universal cover of $BG \vee BH$. In fact it ought to be possible to completely describe the covering theory of $BG \vee BH$ at the level of groupoids, leading to the Kurosh subgroup theorem. For some indication of how this might be done see this blog post, which has some pictures of the special case $G = C_2, H = C_3$.

Source Link
Qiaochu Yuan
Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741

This is not really an answer. The inclusion of homotopy $1$-types into homotopy types has a left adjoint $\tau_{\le 1}$ given by truncation, and hence preserves homotopy limits (e.g. products). The question at hand is whether there is a homotopy-theoretic way to see that it also preserves certain homotopy colimits, namely a certain shape of homotopy pushout.

One might ask more generally what happens when $1$ is replaced by $n$, and here I believe it's not true that the inclusion of homotopy $n$-types preserves homotopy pushouts; e.g. I don't believe it's true that the wedge sum of two homotopy $n$-types is another homotopy $n$-type. (The first counterexample that comes to mind only works rationally: rationally $S^2$ is a $2$-type, but $S^2 \vee S^2$ has rational homotopy in arbitrarily high degrees.)

So there's something special about the inclusion of homotopy $1$-types. At this point you might be happy to learn that it's possible to develop covering space theory for groupoids without any reference to topological spaces. I think this is already enough to prove the desired result, by a construction, at the level of groupoids, of the universal cover of $BG \vee BH$. In fact it ought to be possible to completely describe the covering theory of $BG \vee BH$ at the level of groupoids, leading to the Kurosh subgroup theorem. For some indication of how this might be done see this blog post, which has some pictures of the special case $G = C_2, H = C_3$.