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I know that the following statement is true and I am looking for a reference:

Given a topological category $\mathcal C$ (i.e. morphisms and objects form a space and all maps in the definition of a category are continuous) such that the degeneracy map $s_0: \mathrm{obj}\, \mathcal C \to \mathrm{mor}\, \mathcal C$ is a cofibration. Then all degeneracy maps are cofibrations $s_i: \mathcal N_n \to \mathcal N_{n+1}\mathcal C$ are cofibrations.

This gives a sufficient criteria for $\mathcal N\mathcal C$ to be a good simplicial space, cf. Appendix of G. Segal: CATEGORIES AND COHOMOLOGY THEORIES.

Can anybody give me a reference for that?

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    $\begingroup$ If the source and target maps $s,t: C_1 \to C_0$ are Hurewicz fibrations, then this is a very easy corollary of the main result of R. W. Kieboom, A pullback theorem for cofibrations (1987). $\endgroup$
    – Chris Schommer-Pries
    Commented Sep 18, 2015 at 10:27

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