# Adjointness of Suspension and Shape Path Functors

## TL;DR

This paper explores the adjoint relationship between suspension and shape path functors within a specific subcategory of shape theory, establishing natural bijections and isomorphisms for topological spaces and their shape homotopy groups.

## Contribution

It introduces a subcategory of shape spaces and proves an adjointness property between suspension and shape path functors, linking shape homotopy groups.

## Key findings

- Established a natural bijection between shape morphisms involving suspension and shape path functors.
- Proved isomorphisms between shape homotopy groups of spaces and their shape path spaces.
- Identified a subcategory where these adjointness properties hold.

## Abstract

In this paper, we introduce a subcategory $\widetilde{Sh}_*$ of Sh$_*$ and obtain some results in this subcategory. First we show that there is a natural bijection $Sh (\Sigma (X, x), (Y,y))\cong Sh((X,x),Sh((I, \dot{I}),(Y,y)))$, for every $(Y,y)\in \widetilde{Sh}_*$ and $(X,x)\in Sh_*$. By this fact, we prove that for any pointed topological space $(X,x)$ in $\widetilde{Sh}_*$, $\check{\pi}_n^{top}(X,x)\cong \check{\pi}_{n-k}^{top}(Sh((S^k, *),(X,x)), e_x)$, for all $1\leq k \leq n-1$.