What are important examples of essentially constant systems?

Question

This is the definition of an essentially constant systems.

Definition: :Let $M:I \rightarrow C$ be a filtered diagram. Assume that

(a) exists some $i$ such that $$X \rightarrow M_i \rightarrow X$$ is identity.

(b) For each $j$, exists $k$, and a morphism $M_j \rightarrow M_k$ which factors through $$ M_j \rightarrow X \rightarrow M_i \rightarrow M_k$$

But this all seems a little ad-hoc.

1. What is the big picture of this definition? I suppose the key notion is the hom set commutes with colimits. Somehow relates to compact objects.

2. What are some key examples of essentially constant systems?

Answer 1

The big picture is that a filtered diagram $M$ is essentially constant if it is isomorphic in the category $\text{ind-}C$ to a filtered diagram consisting of the single object $X$ and its identity morphism, i.e. isomorphic to the image of an object $X$ in $C$ under the natural embedding $C\hookrightarrow \text{ind-}C$.

Let me try to motivate / unpack this. Suppose we have two filtered diagrams $M\colon I\to C$ and $N\colon J\to C$, and we want to understand the arrows $\text{colim}M\to \text{colim}N$ which "come from the filtered diagrams". Well, an arrow $\text{colim}M\to \text{colim}N$ is determined by a coherent family of arrows $M_i\to \text{colim}N$ for all $i\in I$. But if we want these arrows to "come from the diagrams", we might reasonably require that each arrow $M_i\to \text{colim} N$ factors through one of the objects $N_j$ in the diagram $N$.

Ok, so for each $i\in I$, we want to look at arrows $M_i\to N_j$ for some $j\in J$. But we should consider two of these arrows to be equivalent if we can see that they induce the same arrow to the colimit, i.e. if they are "eventually the same": $f\colon M_i\to N_j$ and $f'\colon M_i\to N_{j'}$ are equivalent if there are arrows $h\colon N_j\to N_k$ and $h'\colon N_{j'}\to N_k$ in the diagram $N$ such that $h\circ f = h'\circ f$. The filteredness of $N$ implies that this is an equivalence relation. With a bit more thought, you can see that the family of all such equivalence classes of arrows out of $M_i$ is $\text{colim}_{j\in J} \text{Hom}(M_i,N_j)$.

This leads us to consider families $(f_i\colon M_i\to N_{j_i})_{i\in I}$ of (equivalence classes) of arrows to the diagram $N$. Of course, to turn such a family into an arrow $\text{colim}M \to \text{colim}N$, we need to assume the family is coherent with the diagram $M$. So if we have an arrow $h\colon M_i\to M_{i'}$ in the diagram, and $f_i\colon M_i\to N_{j_i}$ and $f_{i'}\colon M_k\to N_{j_{i'}}$ are the corresponding arrows in the family, then we want to assume $f_i$ and $f_{i'}\circ h$ are equivalent in the sense of the last paragraph. With a bit more thought, you can see that the family of all such coherent families of equivalence classes of arrows is $\text{lim}_{i\in I}\text{colim}_{j\in J} \text{Hom}(M_i,N_j)$. Which is exactly the definition of $\text{Hom}(M,N)$, the set of arrows between filtered diagrams in the construction of $\text{ind-}C$.

Next question: When are two filtered diagrams "essentially the same"? Well, it should be when they are isomorphic in the category described above. Unpacking this, an isomorphism $M\cong N$ is witnessed by a coherent family of arrows $(f_i\colon M_i\to N_{j_i})_{i\in I}$ and a coherent family of arrows $(g_j\colon N_j\to M_{i_j})_{j\in J}$ such that the families $(g_{i_j}\circ f_i)_{i\in I}$ and $(f_{i_j}\circ g_j)_{j\in J}$ are equivalent to the families of identity arrows on $M$ and $N$, respectively.

Let's unpack what this means in the case of an isomorphism between a filtered diagram $M\colon I\to C$ and a constant diagram consisting of a single object $X$ and the identity morphism $\text{id}_X$:

1. The family of arrows in one direction is just a single arrow $g\colon X\to M_{i^*}$ for some $i^*\in I$. Coherence is trivial.

2. The family of arrows in the other direction is $(f_i\colon M_i\to X)_{i\in I}$, and coherence says that for all $h\colon M_i\to M_j$ in the diagram, $f_i = f_j\circ h$.

3. In one direction, we require that $f_{i^*}\circ g\colon X\to M_{i^*}\to X$ is the identity map $\text{id}_X$.

4. In the other direction, we require that $(g\circ f_i)_{i\in I}$ is equivalent to the identity family $(\text{id}_{M_i})_{i\in I}$. For all $i\in I$, $g\circ f_i$ is an arrow $M_i\to X\to M_{i^*}$, and $\text{id}_{M_i}$ is an arrow $M_i\to M_i$, and these are equivalent if there is some $k\in I$ and arrows $h\colon M_{i^*}\to M_k$ and $h'\colon M_i\to M_k$ such that $h\circ g\circ f_i$ and $h'\circ \text{id}_{M_i} = h'$ are equal.

So condition (a) in your definition of filtered diagram corresponds to (1) and (3) above, and condition (b) corresponds to (4) above. Condition (2) above comes from the assumption on the stacks project page you link to that $X = \text{colim} M$.

I would prefer to make the definition "$M$ is essentially constant with value $X$" for an arbitrary object $X$, by conditions (1)-(4) above, and then prove that it follows that $X \cong \text{colim} M$.

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I'm not sure I know any particularly interesting examples off the top of my head. One special case that comes up sometimes is when a particular object $X$ is cofinal in a directed diagram, in the sense that for all $i\in I$, there is some $i<j$ in $I$ such that $M_j = X$, and whenever $j<k$ in $I$ and $M_j = M_k = X$, the arrow $M_j\to M_k$ is $\text{id}_X$ (or, more generally, can be composed with a further arrow $M_k\to M_l = X$ to be the identity on $X$). In this situation, it follows that the colimit of the diagram is isomorphic $X$, because the diagram is essentially constant with value $X$.