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Stream: learning: questions

Topic: Cones in a pullback of two categories?

Chris Grossack (they/them) (Sep 12 2024 at 20:21):

Say you have a diagram of $k$-linear stable $\infty$-categories $\mathcal{C} \overset{\pi_\mathcal{C}}{\to} \mathcal{X} \overset{\pi_\mathcal{D}}{\leftarrow} \mathcal{D}$. Then you can form the (homotopy) pullback $\mathcal{C} \times_\mathcal{X} \mathcal{D}$, and you compute (I'm pretty sure):

objects in the pullback are tuples $(c,d, \alpha)$ where $c \in \mathcal{C}$, $d \in \mathcal{D}$, and $\alpha : \pi_\mathcal{C} c \cong \pi_\mathcal{D} d$ is an isomorphism in $\mathcal{X}$.

arrows in the pullback are tuples $(f,g,H) : (c,d,\alpha) \to (c',d',\alpha')$ where $f : c \to c'$, $g : d \to d'$, and $H$ is a homotopy in $\mathcal{X}$ witnessing commutativity of the obvious square (which is hard to typeset in zulip). $H : \alpha' \circ \pi_\mathcal{C}f \sim \pi_\mathcal{D} g \circ \alpha$.

Chris Grossack (they/them) (Sep 12 2024 at 20:23):

Now for the question: Is there a convenient way to express $\text{Cone}(f,g,H)$ in terms of $\text{Cone}(f)$, $\text{Cone}(g)$, and something to do with the two cell $H$?

Chris Grossack (they/them) (Sep 12 2024 at 20:23):

If there's a way to do this expressed in the language of (pretriangulated) dg or $A_\infty$ categories rather than in the language of stable $\infty$-categories, that's fine too.

Kevin Carlson (Sep 12 2024 at 20:49):

Since the arrow category commutes with the pullback and the 2-functor of taking pullback preserves adjunctions, the left adjoint to the insertion-of-zero functor $C\times_X D \to (C\times_X D)^\to= C^\to \times_{X^\to} D^\to$ will be given by the pullback of the analogous left adjoints on $C,X,$ and $D.$ In other words to take the cone on a morphism $(f,g,H)$ you take the cones of $f,g$ in $C,D,$ and then take the cone on $H$ as well, seen as a morphism in $X^\to,$ using that the cone is a functor $X^\to \to X.$

Kevin Carlson (Sep 12 2024 at 20:51):

As far as formalities go, the 2-functor I'm referring to is happening in the homotopy 2-category of $\infty$-categories, but I probably wouldn't worry too precisely about that until you've decided whether you feel good about the concrete answer at the end.

Chris Grossack (they/them) (Sep 13 2024 at 20:28):

Thanks! This is super helpful. Do you mind saying more about precisely what the "insertion-of-zero" functor is? I can see how this is probably related to the cone, and I could probably do it myself but I'm working on something else right now and it's easier to ask, haha.

Chris Grossack (they/them) (Sep 13 2024 at 20:29):

But if I understand, you're saying that cone as a functor $X^\to \to X$ is left adjoint to this "insertion of zero" functor $X \to X^\to$, right?