In homotopy theory, we can construct the $\infty$-category of spaces from the ordinary category of oriented manifolds $\rm Man$, by freely co-completing it, imposing gluing relations, and homotopy invariance. This sort of procedure is described model-categorically by Dugger in https://arxiv.org/abs/math/0007070.
My question is: Is it possible to perform an analog of this construction, where instead of homotopy invariance, we only impose that $\mathbb R^1$ is invertible? That is, we freely make $\mathbb R^1 \times -$ an equivalence.
Is there a source for this sort of localization of multiplication by objects in a symmetric monoidal category? It is analogous to the process of constructing spectra.
If this sort of localization exists, what do we get in this case? I suppose it's possible that we just get the ordinary homotopy category again. Alternately, we could get something related to the proper homotopy category.