Levyts'ka V. Algebraic-topological properties of functors generated by functional spaces

A notion of functorial topologization of the sets of continuous maps is introduced and it is proved that there is no regular functorial topologization with the Hilbert space in bounded-weak topology as one of its values. A functorial topologization for the set of piecewise-linear maps which transforms compact infinite polihedra to infinite dimensional manifolds is constructed. A notion of functorial differentiable structure on infinite dimensional manifolds modeled on topological linear spaces is introduced and it is shown that there is no such a structure on the space of continuous maps into a space that doesn't have a structure of differentiable (finite-dimensional) manifold. The rest of results concerns properties of the contravariant functor Cp of continuous functions in the topology of pointwise convergence acting in the category TYCH of Tychonov spaces and continuous maps. Recall that monad T=(T,h,m) on a category C consist of an endofunctor T:C®C and natural transformations h:1c®T (unity) and m : T2®T (multiplication) that satisfy the conditions of associativity and two-side unity. The Kleisli category CT of the monad T is the category whose class of objects coinsides with that of C and whose morphisms are T-valued morphisms of C with naturally defined operation. A criterion is established for existence of extensions of contravariant functors onto the Kleisli categories of monads. Namely, it is proved that there exists a bijective equivalence between extensions of contravariant functor S onto the category CT and natural transformations x: F®TFT satisfying the conditions TFh°x=hF and TFm°x=mFT2°TxT°x. This criterion is applied to the problem of extension of the contravariant functor Cp onto the Kleisli category of monad in the category of Tychonov spaces generated by the functor Cp2 and also the hyperspace monad and its submonad of the hyperspace of finite sets. In particular, it is proved that the contravariant functor Cp of spaces of functions in the pointwise-convergence topology, as well as all i ts (transfinite) iterations, has an extension onto the Kleisli category of the monad (Cp2 , h, m). Sufficient conditios are given for existence of extensions of compositions of contravariant functors onto the Kleisli categories. A general criterion for existence of lifting of contravariant functors onto the categories of T-algebras, which is based on existence of natural transformations of special form, is established. This criterion is applied to the contravariant functor Cp and the monad generated by the functor Cp2 and it is proved that there exists a lifting of the functor Cp on the category of T-algebras for the monad T= (Cp2 , h, m). An analogous result is also proved for the contravariant functor Bp(a) of real-valued functions of Baire class a in the topology of pointwise convergence.