A proset structure induced from homotopy classes of maps and a classification of fibrations
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Firstly we consider preorders (not necessarily partial orders) on a canonical quotient of the set of the homotopy classes of continuous maps between two spaces induced by a certain equivalence relation $ \sim_{{\mathcal E}R}$. Secondly we apply it {to} a classification of orientable fibrations over $Y$ with fibre $X$. In the classification theorem of J. Stasheff \cite{Sta} and G. Allaud \cite{All}, they use the set $[Y,B{\rm aut}_1X]$ of homotopy classes of continuous maps from $Y$ to $B{\rm aut}_1X$, which is the classifying space for fibrations with fibre $X$ due to A. Dold and R. Lashof \cite{DL}. In this paper we give a classification of fibrations using a preordered set (abbr., proset) structure induced by $[Y,B{\rm aut}_1X]_{{\mathcal E}R}:=[Y,B{\rm aut}_1X]/\sim_{{\mathcal E}R}$.
Abstract
Firstly we consider preorders (not necessarily partial orders) on a canonical quotient of the set of the homotopy classes of continuous maps between two spaces induced by a certain equivalence relation $ \sim_{{\mathcal E}R}$. Secondly we apply it {to} a classification of orientable fibrations over $Y$ with fibre $X$. In the classification theorem of J. Stasheff \cite{Sta} and G. Allaud \cite{All}, they use the set $[Y,B{\rm aut}_1X]$ of homotopy classes of continuous maps from $Y$ to $B{\rm aut}_1X$, which is the classifying space for fibrations with fibre $X$ due to A. Dold and R. Lashof \cite{DL}. In this paper we give a classification of fibrations using a preordered set (abbr., proset) structure induced by $[Y,B{\rm aut}_1X]_{{\mathcal E}R}:=[Y,B{\rm aut}_1X]/\sim_{{\mathcal E}R}$.
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