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As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ (Let's assume everything is CGWH.) to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

Up here, no point-set topological restrictions are needed, it even all works if the spaces are not compactly generated.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. In the case of well pointed compactly generated groups, their simplicial nerves will be "good" in the sense of Segal. A survey on the results of that manner is given here.

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ (Let's assume everything is CGWH.) to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. A survey on the results of that manner is given here.

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

Up here, no point-set topological restrictions are needed, it even all works if the spaces are not compactly generated.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. In the case of well pointed compactly generated groups, their simplicial nerves will be "good" in the sense of Segal. A survey on the results of that manner is given here.

deleted 117 characters in body
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archipelago
archipelago
  • 3k
  • 21
  • 37

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ (Let's assume everything is CGWH.) to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) This yields the classical classifying space of $G$ defined by Milnor as Segal remarked in the paper just cited. A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. A survey on the results of that manner is given here.

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ (Let's assume everything is CGWH.) to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) This yields the classical classifying space of $G$ defined by Milnor as Segal remarked in the paper just cited. A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. A survey on the results of that manner is given here.

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ (Let's assume everything is CGWH.) to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. A survey on the results of that manner is given here.

Source Link
archipelago
archipelago
  • 3k
  • 21
  • 37

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses.

Let me present one case for which the question can be answered positive which is basically the case Dan Ramras mentioned.

We define $BG$ for a topological group $G$ (Let's assume everything is CGWH.) to be the fat realization of the simplicial space obtained by applying the topological nerve construction to the topological category one obtains by regarding $G$ as a category in the usual way and including its topology. (See Segal's 'Classifying Spaces and Spectral Sequences' §3) This yields the classical classifying space of $G$ defined by Milnor as Segal remarked in the paper just cited. A continuous group homomorphism $G\rightarrow H$ which is a homotopy equivalence will induce a morphism of the corresponding simplicial spaces which is a degreewise homotopy equivalence. (This is easy to check, right from the definitions.) Since we have chosen the fat realization the induced map $BG\rightarrow BH$ will be a homotopy equivalence by Proposition A.1 in Appendix A in Segal's 'Catgeories and Cohomology Theories'.

In the last paper cited, you'll also find information about the question, in which cases the fat realization is homotopy equivalent to the usual one. A survey on the results of that manner is given here.