# Does the fat realization of simplicial spaces commute with finite limits up to homotopy?

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.

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