What is the meaning of the following condition on a morphism in sSets or simplicial topological spaces:
a morphism becomes a direct product after the pullback along the shift(decalage) morphism ?
Is it related to being locally trivial ? Is such a morphism necessarily a fibration (say provided the base is fibrant; I understand the decalage morphism is a fibration under this assumption.) ?
Let me explain this in notation. Let $[+1]:\Delta\to \Delta$ be the endomorphism of $\Delta$, $[+1](n):=1+n$ adding a new least element, and for a simplicial set $B$ let $B\circ [+1] \to B$ be the shift(decalage) morphism "forgetting the first coordinate". Let $f:X\to B$ be a morphism in sSets.
What is the meaning of the condition that $B\circ [+1]\times_B X \to B\circ [+1]$ is a direct product, i.e. there is $F$ and an isomorphism $B\circ [+1]\times F \approx B\circ[+1]\times_B X$ over $B\circ [+1]$?
I understand that if $B$ is fibrant, then $B\circ [+1]\to B$ is a fibration, and if $F$ is fibrant, the map $B\circ[+1]\times F\to B\circ [+1]$ is a fibration, and so perhaps $X\to B$ would also be a fibration in this case ?
Any references would be appreciated.