The n-Category Café

Merely having each triangle commute up to a specified homotopy is not very much… I can’t imagine getting your hoped-for result from that condition.

Sure, I was just thinking of a simple pushout diagram. For that case of “very small colimit diagrams” this page 15 in Homotopy theory of diagrams which I mentioned gives an easily applied prescription, whereas the general case, according to this article, is way more involved.

I have just a morphism $p : C \to D$ of $\omega$-groups and can compute in various low dimensional special cases that its ordinary kernel $ker(p) \to C$, the pullback along the map from the point, has not just the usual property you would expect from an ordinary kernel, but the property that

given a morphism $A \to C$ such that $A \to C \stackrel{p}{\to} D$ is just homtopic to the map that factors through the point, then already it lifts, up to homotopy, to a map that factors through $\mathrm{ker}(p)$.

I can do this computation explicitly using transformations, modifications, etc, of $n$-functors, which quickly gets a bit out of control as $n$ grows.

Now, I suspect that the model structure comes to the rescue: namely I can easily show using that prescription on that p. 15 that in the special case that I am looking at the ordinary kernel $ker(p)$ is already weakly equivalent to the homotopy hernel $hoker(p)$.

So, if on top of all this which I already know the statement holds which I expect but don’t know, namely that “homotopy coherent cones”, as I take it they are to be called, have up to homotopy unique maps into their tip out of the homotopy colimit, then that would not only sumarize the partial results I have but also give the expected statement for all $n$.

Then we have a ‘homotopy coherent’ diagram, and these are nicely related to homotopy limits and colimits.

Good. That’s what i am expecting. i just didn’t know where to find the precise statement. Thanks for providing some! I’ll have a look.

John,
Thanks for the plug!

I would suggest that for homotopy limits and colimits it is also worth looking at the paper by Bourn and Cordier on indexed limits.

D. Bourn and J.-M. Cordier, A general formulation of homotopy limits , J. Pure Appl. Algebra, 29, (1983), 129–141.

After having taken care of a couple of duties, I am now coming back to this discussion here.

Mike Shulman kindly wrote:

you may find section 10 of this paper [Shulman: Homotopy limits and colimits and enriched homotopy theory] helpful.

Yes, indeed, I am finding this quite helpful. In particular, I find in this the general statement that I was looking for, on p. 29:

We […] make precise the notion of “homotopy coherent cones” […]. A homotopy coherent cone under a diagram $F$ should consist of a vertex $C$ and a coherent transformation $F \sim\to G$, where by $C$ we mean the diagram constant at the vertex $C$. […] Therefore, a representing object for homotopy coherent cones under $F$ is precisely […] the homotopy colimit of $F$. Thus the latter really does have a “homotopical universal property”.

Now I need to go back to the previous sections to unwrap this.

While I am busy learning this stuff, maybe somebody can help me by just saying whether or not the answer to my concrete problem on a simple special case is Yes or No:

Consider an ordinary pushout cone $$\array{ X &\to& B \\ \downarrow && \downarrow \\ A &\to& col }$$ in some category. $col$ indicating the ordinary colimit. Now assume there is some model category structure around and that it so happens that with respect to that $col$ happens to be weakly equivalent to the homotopy colimit

$$hocolim \left( \array{ X &\to& B \\ \downarrow && \\ A } \right) \stackrel{\simeq}{\to} col \,.$$

Assume in this case furthermore that we have morphisms $$\array{ && B \\ &&\downarrow \\ A &\to& V }$$ which do not quite make a cone, but make a “homotopy coherent cone” in that the outermost boundaries of the following diagram

are homotopic - as indicated by the long double arrow.

Then: is it true or not that this alone is sufficient and necessary for the indicated dashed arrow to exist, such that it makes the indicated triangles commute up to homotopy, as indicated by the two short double arrows?

I have another question on Mike Shulman’s Homotopy limits and colimits and enriched homotopy theory:

When it comes to enriched homotopy theory from section 11 on, there is, stated on page 34, the assumption that the category $V$ that one is enriching over is related by a strong monoidal adjunction $V \leftrightarrow SimpSet$ to simplicial sets.

I suppose for $V := \omega Cat$, the $\omega$-nerve $$N : \omega Cat \to SimpSet$$ does satisfy the required conditions?

(In the sense of the discussion on page 33 this comes from the canonical functor $O(\Delta^{(-)}) : \Delta \to \omega Cat$ provided by the orientals.)

If this is correct, that $V = \omega Cat$ satisfies the assumptions in section 13, would this generalize to $\omega$-categories internal to any category of sheaves?