It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. However, I know next to nothing about the model structure on CGWH spaces.


Questions:
What are the analogs of Kan fibrations, trivial fibrations, cofibrations, and weak cofibrations in CGWH?


Are there analogs of left, right, inner fibrations in CGWH (along with their corresponding anodyne maps)?



Analogs of left, right, inner, and Kan fibrations in CGWH
The analogs are:


* Serre fibrations (map with lifting property with respect to $I^n\times 0\to I^{n+1}$)
* trivial Serre fibrations (map with lifting property with respect to $S^{n-1}\to D^n$)
* retracts of maps built by attaching $S^{n-1}\to D^n$
* retracts of maps built by attaching $I^n\times 0\to I^{n+1}$.


There are any number of references to the Quillen model structure on Top, starting with Quillen's book; you can google for the article by Dwyer and Spalinksi on "Homotopy Theories and Model Categories". 


Hovey's book "Model Categories" also does this, and gives significant attention to the cases of k-spaces and CGWH; the description of the fibrations/cofibrations is the same in these categories, but some care is needed to make sure the proof that you get a model category goes through. Hirschhorn's book may also do this, though somebody seems to have my copy, so I can't check.


I can't imagine you can define left/right/inner fibrations in Top or CGWH. Topologically, an n-simplex has no "left" or "right"; you always have a homeomorphism $\Delta^n\to \Delta^n$ which permutes the vertices however you please. 


