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I'm willing to work in the category of compactly generated Hausdorff spaces.

The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, hence commute with arbitrary finite limits. (A reference with a proof is Corollary 11.6 in May's 'Geometry of Iterated Loop Space')

It is stated in various places (Proposition 9 on this page of the nLab, Remark 2.23 of 'Homotopy Theory of Orbispaces' by Gepner and Henriques) that the fat geometric realization preserves finite limits up to homotopy, but I could not find a solid reference for a proof of this statement. Since it is obvious that it preserves the terminal object up to homotopy, it suffices to have a proof for the pullback-case.

If all simplicial spaces in consideration are 'good' then the statement follows from the preservation for the ordinary realization, but I'm interested in the general case.

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    $\begingroup$ I don't quite understand the vote to close. $\endgroup$ – Todd Trimble Jul 10 '15 at 15:46
  • $\begingroup$ Note that fat realisation preserves pullbacks (but poss with k-topology), just not terminal obj; it gets sent to Δ^\infty $\endgroup$ – David Roberts Jul 10 '15 at 15:46
  • $\begingroup$ I do not understand your comment. The fat realization does not even preserve products. $\endgroup$ – Simp Jul 10 '15 at 15:51
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    $\begingroup$ Products are pullbacks of X→ 1←Y, and 1 but not preserved. More generally, if I'm not wrong, B' should preserve finite connected limits. $\endgroup$ – David Roberts Jul 11 '15 at 4:10
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    $\begingroup$ It is stated (without a proof or a reference) as Remark 2.23 in "Homotopy Theory of Orbispaces" by David Gepner and @AndréHenriques . $\endgroup$ – Simp Jul 13 '15 at 18:00

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