@PiotrPstrągowski is it somehow obvious from the description you gave of this loops functor, or of Lurie's description of it, that it has a left adjoint?

@JonathanBeardsley As a functor from pointed simplicial sets to simplicial sets, $\Omega$ has a left adjoint which sends a simplicial set $A$ to the pushout of the span $\Delta^0 \star A \leftarrow A \rightarrow \Delta^0$ (which you might write as $(\Delta^0 \star A)/A$). This explicit description follows from the definition of $\Omega(X,x) = Hom_X^L(x,x)$ as the fibre of $x \backslash X \rightarrow X$ over $x$ and the usual join--slice adjunction.

(N.B. I write $x \backslash X$ where Lurie writes $X_{x/}$.)

Actually that's not quite right, you have to consider $\Omega$ as a functor from pointed simplicial sets to pointed simplicial sets to get a left adjoint.

The left adjoint sends a pointed simplicial set $(A,a)$ to the pushout of the span $\Delta^0 \star A \leftarrow (\Delta^0 \star \{a\}) \cup_{\{a\}} A \rightarrow \Delta^0$. That is, the quotient of $\Delta^0 \star A$ where we crush the subcomplex $(\Delta^0 \star \{a\})\cup A$ to a point, which becomes the distinguished basepoint.

here's a crazy theorem i just learned. let M be a smooth compact manifold of dimension $\geq 3$. choose any smooth function f on M that is somewhere negative. then, there exists a riemannian metric on M such that f is its scalar curvature. (this explains why "positive scalar curvature" is so well-studied.)

Is there any decent notion of "integral extension" or "integral closure" for commutative ring spectra. (I'm guessing there isn't ...)

@AlexanderCampbell oh wow as always thankyou Alexander

So I suppose this is sort of a biased suspension in a way.