着想 #197

Internalizing knot spaces in SDG # Internalizing knot spaces in SDG ## Core idea The usual knot space \[ \mathrm{Emb}(S^1,\mathbb R^3) \] may be recoverable as an internal object of a smooth topos, not by externally imposing both injectivity and immersion, but by interpreting the naive internal formula \[ \{f:S^1\to\mathbb R^3 \mid \forall x,y:S^1,\ f(x)=f(y)\Rightarrow x=y \} \] inside a Synthetic Differential Geometry model such as a Cahiers-type topos. The point is that the same formula has different force depending on the available generalized points. ## Contrast with ordinary smooth sets In smooth sets or diffeological spaces, the internal injectivity condition only says that a smooth family \[ F:U\times S^1\to\mathbb R^3 \] has pointwise injective fibres \[ F_u:S^1\to\mathbb R^3. \] This does not force the immersion condition. For example, \[ f(e^{it})=(\sin^3 t,\cos t,0) \] is injective but has zero derivative at \(t=0,\pi\), so it is not a smooth embedding. Thus, in ordinary smooth-set or diffeological semantics, \[ \mathrm{Inj}(S^1,\mathbb R^3) \] is too large to be the knot space. ## SDG insight In a Cahiers-type topos, the universal quantifiers over \(S^1\) range over infinitesimal generalized points as well as ordinary points. If \(f:M\to N\) has a critical direction, say \[ df_p(v)=0 \] for some nonzero tangent vector \(v\in T_pM\), then an infinitesimal point \[ x(d)=p+dv \] cannot be separated from \[ y(d)=p \] by \(f\), because \[ f(p+dv)=f(p)+d\,df_p(v)=f(p). \] Therefore internal injectivity in SDG detects not only ordinary injectivity but also infinitesimal separation, i.e. immersion. For \(S^1\to\mathbb R^3\), injective immersion is equivalent to smooth embedding because \(S^1\) is compact and \(\mathbb R^3\) is Hausdorff. Hence the expected slogan is: \[ \mathrm{Inj}_{\mathrm{Cahiers}}(S^1,\mathbb R^3) \quad\text{recovers}\quad \mathrm{Emb}(S^1,\mathbb R^3) \] at ordinary stages, while giving a formal infinitesimal enhancement internally. ## Possible research direction Study the object \[ \mathrm{Inj}_{\mathcal E}(S^1,\mathbb R^3) \] in a well-adapted SDG topos \(\mathcal E\), and compare its ordinary reflection with the classical embedding space of knots. Questions: 1. For which SDG models does internal injectivity of ordinary manifolds coincide with injective immersion? 2. Does the internal object give a useful formal thickening of the classical knot space? 3. Can isotopy, knot invariants, or Vassiliev-type discriminants be expressed internally? 4. Can this viewpoint relate embedding calculus, diffeological knot spaces, and SDG? 5. Is there a clean internal definition of the quotient \[ \mathrm{Emb}(S^1,\mathbb R^3)/\mathrm{Diff}(S^1) \] as the space of unparametrized knots? ## Caveat This is currently a speculative packaging of standard ingredients: smooth mapping spaces, internal logic, diffeological spaces, and infinitesimal tests in SDG. The key model-dependent claim to verify is that Cahiers-internal injectivity of ordinary maps is exactly injective immersion.