So here's a possibly easy/dumb question: given a Kan complex $X$ and a sub-Kan-complex $Y\subset X$, and a functor of quasicategories $F:X\to C$, we can take the colimit of the composition $Y\to X\overset{F}\to C$. This should still determine a diagram $X\to C$ right? What's the indexing simplicial set of this new diagram with part of the colimit already taken?

Why should it determine a diagram?

In other words, there should be a map $X\to C$ where the everything in $Y$ has been mapped to the colimit of that composition.

Hm...

Are you referring to the left Kan extension along $X\to X/Y$?

Well.... sort of.

I mean... that's the type of thing I'm thinking about.

What is X/Y there? Just the cofiber?

Yes

Right... hrm.

@DenisNardin sent u an email

@JonBeardsley are you getting at $X \coprod_Y Y^\triangleright$?

let X, Y be pointed spaces, let [X, Y] be the space of maps X -> Y, and let [X, Y]_{\ast} be the space of pointed maps X -> Y. what's the homotopy fiber of the natural map [X, Y]_{\ast} \to [X, Y]? I think it's Omega Y... does that sound right?

@Qiaochu yep. Map_*(X,Y) is the fiber of Map(X,Y) -> Map({x},Y) = Y over the basepoint, and so the next fiber is the loop space

@Tyler: ah, that's very nice! thanks

The homology of the cubical construction of an abelian group A gives you the homology of the corresponding Eilenberg-Mac Lane spectrum. Is there a construction which would somehow give the unstable homology of the Eilenberg-Mac Lane spaces?

math.jhu.edu/~wsw/papers2/math/39-hopf-rings-expo-2000.pdf might be helpful, especially the discussion starting at the bottom of page 8. The main keyword here is "Hopf ring"

thanks, I'm aware of that

it gives a compact and clear description of them, for sure, but it's not obtained from something of different nature done algebraically to the abelian group

fair enough

@AaronMazel-Gee yeah i think so. but that's just the cone over $Y$. I want to do this in a more structured way, and I think I need the operadic Kan extension framework.

@JonBeardsley well, this isn't a kan complex anymore -- it's got non-invertible arrows from $y \in Y$ to $\infty \in \triangleright$. you might call it a "directed" cofiber. anyways, i'm not sure what you mean by "more structured" here, if X and Y are just kan complexes

@AaronMazel-Gee ah that's a good point. i guess i hadn't really thought about the fact that it wasn't a kan complex. yeah... hm. and ultimately i worked it out, but by more structured i mean being n-fold loop spaces.

Hey all, in Remark 3.1.3.15 of Higher Algebra, Lurie asks for a certain morphism $h'':M^\otimes\to C^\otimes\times\Delta^1$ to be an operadic q-left Kan extension. However, his usual notation for such things is $M^\otimes\to C^\otimes$. Is it possible he means something like $h''$ composed with projection onto $C^\otimes\times\{1\}$ or something?

The reason is, the set-up is really about extending along a map of $\infty$-operads $A^\otimes\to B^\otimes$, given a map $A^\otimes\to C^\otimes$. So one should expect the Kan extension to land in $C^\otimes$ rather than $C^\otimes\times\Delta^1$... maybe?

In other words, it doesn't quite look like Definition 3.1.2.2.