Abstract: One of the longstanding problems in homotopy theory is the question how, for a given space A, one can characterize the class of spaces which are homotopy equivalent to the pointed mapping spaces Map_* (A,Y). In case where A is an n-dimensional sphere S^n this problem was solved in several ways using the machinery of categorical algebra: operads, PROPs, Segal special Delta-spaces, etc. The common feature of all these descriptions is that they detect if a given space X is of a type of a mapping space from S^n using only certain maps between finite products of X. This shows that the mapping spaces Map_* (S^n,Y) are essentially algebraic objects. The talk will describe how one can try to generalize this approach to describe mapping spaces for spaces A other than S^n and the obstructions that one encounters.

This is a joint project with B. Badzioch.