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Let $\mathcal T$ be a category of "nice" topological spaces (CW?) and continuous maps between them. We can construct the homotopy category $\mathrm{Ho}\mathcal{T}$ which gives for any two objects $X,Y$ the ($\infty$-groupoid of) homotopy classes between them $[X,Y]$.

Now hopefully restricting the morphisms of the previous category to embeddings (regular moonics, i.e. morphisms that are equalisers) we can get some sort of (sub)category of embeddings $\mathsf{Emb}(\mathcal{T})$ whose model structure can maybe be inherited giving a homotopy category $\mathrm{Ho}\mathsf{Emb}(\mathcal{T})$, where homotopy classes are really just isotopy classes.

Picking $S^1$ and $S^3$ from $\mathrm{Ho}\mathsf{Emb}(\mathcal{T})$ we get a category of knots $[S^1,S^3]$ which kind of sucks as they are all isotopic so its a pretty useless bunch.

Now I want to construct a category whose objects are knots, and morphisms are ambient isotopy classes. This would allow me to do something similar as above and actually get some interesting information out of this. However I am having a lot of trouble interpreting knot theory in this way:

  1. This could be because there is something intrinsic about knot theory that makes it unsuseptable to categorical thinking, which would indicate inadaquacy of categorical lanugage in this instance. (I highly doubt this, I just don't think I am thinking hard enough).

  2. I don't really fully understand knot theory.

I am going to assume it is (2.) that is my problem and hope that this question can help clear things up.

Is there a categorical/homotopy theoretic approach to knot theory like I outlined above? If so where can I find the literature?

I am essentially looking for a formulation of knot theory that looks like it came from nLab.

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    $\begingroup$ The smooth embeddings of $S^1$ into $S^3$ are parameterized by a topological space. Its fundamental groupoid is probably your desired category. $\endgroup$ – Will Sawin Jul 6 '18 at 11:37
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    $\begingroup$ While I agree with @WillSawin about which category you are looking for, I want to express a great deal of skepticism that you will be able to use $\infty$-categorical thinking to prove serious results about knot theory. Just because you have a hammer doesn't mean everything is a nail... $\endgroup$ – Andy Putman Jul 6 '18 at 20:41

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