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.
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.