# What is this analog of $\mathbb A^1$ / proper homotopy theory?

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.

• Are you considering the real line as a manifold or as a topological space? In the latter case, there's nothing new. Dec 1, 2018 at 0:39
• @user40276 Currently as a manifold. Do you expect these cases to be different? This question is for "curiosity only," so anything relevant you can say / pointers you have in the direction of more info are welcome. Dec 1, 2018 at 3:19
• I think that if you invert the line in the category of $\infty$-sheaves over cartesian spaces (which is equivalent to $\infty$-stacks over manifolds) you will get the category of spaces back (you will get stacks over a point). A more interesting thing would be to invert the circle as a differentiable manifold to get a motivic differentiable theory. But maybe the result will again be trivial. Maybe you're interested in something like the property of exhibiting cohesion as in ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos Dec 1, 2018 at 3:34
• @user40276 OP is asking specifically NOT about localizing at X\times R^1 -> X (which is addressed in the first paragraph of the question) but instead about forcing the object R^1 to become invertible with respect to Cartesian product (formally adding an object R^{-1} to the category). This is possible using a construction in Mario Robalo’s thesis but it’s not a localization, your objects are gonna be some kind of spectra wrt R^1. I don’t think you’d recover any well known construction, it just looks like a weird thing to do
– user123627
Dec 1, 2018 at 10:09
• @Gasterbiter Thanks for the reference--- Marco's thesis has a lot of useful info. I'd argue that for geometers (who know about things like dimension) this construction is not so weird. For instance people often take the grothendieck ring of varieties, and they invert $\mathbb A^1$-- they don't set it equal to $1$. Dec 1, 2018 at 15:46