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The geometric realisation functor (read: homotopy colimit for nice situations) from simplicial spaces to $Top$ preserves pullbacks only when you take the $k$-ification of the product in $Top$, or work with compactly generated spaces (Edit: or a convenient category of spaces). This is false in the category of all spaces with the ordinary product. See e.g. here in the nLab. |
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The geometric realisation functor (read: homotopy colimit for nice situations) from simplicial spaces to $Top$ preserves pullbacks only when you take the $k$-ification of the product in $Top$, or work with compactly generated spaces. This is false in the category of all spaces with the ordinary product. See e.g. here in the nLab. |
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