Monomorphisms of diagrams in an $\infty$-category

Question

Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions:

1. If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a monomorphism in $\mathcal{C}$ at every component, is it a monomorphism in $Fun(K,\mathcal{C})$?
2. If every 1-simplex in the image of $f$ is a monomorphism, is it true that the maps $lim(f)\to f(\sigma)$ are monomorphisms for all 0-simplices $\sigma\in K$?

I don't know if these are going to be true at this level of generality. Ultimately I'm interested in the very special case of $\mathcal{C}=Gpd_\infty$ and $K$ being the subcategory of $\Delta$ spanned by injections (i.e. semicosimplicial objects), so if there's a simpler answer in that case I'd be very happy to know it.

Answer 1

For completeness, and because I cannot figure out the general case (cf. my comment below Daniel's answer), let me prove the following: if $f:\Delta\to C$ is a functor which, when restricted to $\Delta_{inj}$ takes values in monomorphisms, then $\lim_\Delta f\to f([0])$ is a monomorphism.

This is simply the following observation : $f([0])\simeq \lim_\Delta f(-*[0])$ (as $f(-*[0])$ is a split-augmented simplicial object), and along thsi identification, the projection map corresponds to $\lim_\Delta (f\to f(-*[0]))$. Now each $[n]\to [n]*[0]$ is an injection, and so this map is $\lim_\Delta$ of monomorphisms, and hence is a monomophism itself.

This seems to use something specific about the combinatorics of $\Delta$.

When $K$ is general, and say has a final object $\infty$, then one can argue similarly: one has a composite $\lim_K f\to f(k) \to f(\infty)$, and thus by some version of 2-out-of-3 it suffices to prove that $\lim_K f\to f(\infty)$ is a monomorphism, but this follows from it simply being $\lim_K (f\to f(\infty))$. My attempt for a general $K$ with $|K|\simeq *$ was to try and prove that the inclusion $K\to K^\triangleright$, where we add a final object, preserves the property of every map being a mono, i.e. that for every $k$, $f(k)\to \mathrm{colim}_K f$ would be a monomorphism - this step is where one would use $|K|\simeq *$, but I didn't manage to prove it.

There is a somewhat easy-to-prove version which is when $K$ is weakly contractible via a zig-zag of functors : if there is a zigzag $id_K \leftarrow h_1 \to h_2 \leftarrow \dots \to h_n$ where $h_n$ is a constant functor, then one can do a similar argument to the one for $\Delta$ and prove the result. I don't remember exactly how often we can expect this: is it the case that any weakly contractible $K$ is a filtered colimit of $K$'s admitting such a zigzag ?

Answer 2

(Assuming $\mathcal{C}$ is finitely complete for convenience)

A partial answer at least:

For the first, $\eta$ is an monomorphism iff its diagonal map $\delta : f \to f \times_g f$ is an isomorphism. But we can check if $\delta$ is an isomorphism by checking each of its components and, since limits are computed pointwise, the component at $k : K$ is $f(k) \to f(k) \times_{g(k)} f(k)$. So this is equivalent to asking that each component of $\eta$ be a monomorphism. (In fact, this seems to prove the condition to be both necessary and sufficient)

For the second, I think as written binary products ($K = \partial \Delta^1$) give a counter-example: any diagram satisfies the condition (there are no non-degenerate 1-cells) but $A \times B \to A$ is hardly ever a monomorphism. I'm not sure about the case where $K = \Delta$ though.