## 4.22 Essentially constant systems

Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram in a category $\mathcal{C}$. Assume the index category $\mathcal{I}$ is filtered. In this case there are three successively stronger notions which pick out an object $X$ of $\mathcal{C}$. The first is just

\[ X = \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} M_ i. \]

Then $X$ comes equipped with the coprojections $M_ i \to X$. A stronger condition would be to require that $X$ is the colimit and that there exist an $i \in \mathcal{I}$ and a morphism $X \to M_ i$ such that the composition $X \to M_ i \to X$ is $\text{id}_ X$. An even stronger condition is the following.

Definition 4.22.1. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram in a category $\mathcal{C}$.

Assume the index category $\mathcal{I}$ is filtered and let $(X, \{ M_ i \to X\} _ i)$ be a cocone for $M$, see Remark 4.14.5. We say $M$ is *essentially constant* with *value* $X$ if there exist an $i \in \mathcal{I}$ and a morphism $X \to M_ i$ such that

$X \to M_ i \to X$ is $\text{id}_ X$, and

for all $j$ there exist $k$ and morphisms $i \to k$ and $j \to k$ such that the morphism $M_ j \to M_ k$ equals the composition $M_ j \to X \to M_ i \to M_ k$.

Assume the index category $\mathcal{I}$ is cofiltered and let $(X, \{ X \to M_ i\} _ i)$ be a cone for $M$, see Remark 4.14.5. We say $M$ is *essentially constant* with *value* $X$ if there exist an $i \in \mathcal{I}$ and a morphism $M_ i \to X$ such that

$X \to M_ i \to X$ is $\text{id}_ X$, and

for all $j$ there exist $k$ and morphisms $k \to i$ and $k \to j$ such that the morphism $M_ k \to M_ j$ equals the composition $M_ k \to M_ i \to X \to M_ j$.

Please keep in mind Lemma 4.22.3 when using this definition.

Which of the two versions is meant will be clear from context. If there is any confusion we will distinguish between these by saying that the first version means $M$ is essentially constant as an *ind-object*, and in the second case we will say it is essentially constant as a *pro-object*. This terminology is further explained in Remarks 4.22.4 and 4.22.5. In fact we will often use the terminology “essentially constant system” which formally speaking is only defined for systems over directed sets.

Definition 4.22.2. Let $\mathcal{C}$ be a category. A directed system $(M_ i, f_{ii'})$ is an *essentially constant system* if $M$ viewed as a functor $I \to \mathcal{C}$ defines an essentially constant diagram. A directed inverse system $(M_ i, f_{ii'})$ is an *essentially constant inverse system* if $M$ viewed as a functor $I^{opp} \to \mathcal{C}$ defines an essentially constant inverse diagram.

If $(M_ i, f_{ii'})$ is an essentially constant system and the morphisms $f_{ii'}$ are monomorphisms, then for all $i \leq i'$ sufficiently large the morphisms $f_{ii'}$ are isomorphisms. On the other hand, consider the system

\[ \mathbf{Z}^2 \to \mathbf{Z}^2 \to \mathbf{Z}^2 \to \ldots \]

with maps given by $(a, b) \mapsto (a + b, 0)$. This system is essentially constant with value $\mathbf{Z}$ but every transition map has a kernel.

Here is an example of a system which is not essentially constant. Let $M = \bigoplus _{n \geq 0} \mathbf{Z}$ and to let $S : M \to M$ be the shift operator $(a_0, a_1, \ldots ) \mapsto (a_1, a_2, \ldots )$. In this case the system $M \to M \to M \to \ldots $ with transition maps $S$ has colimit $0$ and the composition $0 \to M \to 0$ is the identity, but the system is not essentially constant.

The following lemma is a sanity check.

Lemma 4.22.3. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. If $\mathcal{I}$ is filtered and $M$ is essentially constant as an ind-object, then $X = \mathop{\mathrm{colim}}\nolimits M_ i$ exists and $M$ is essentially constant with value $X$. If $\mathcal{I}$ is cofiltered and $M$ is essentially constant as a pro-object, then $X = \mathop{\mathrm{lim}}\nolimits M_ i$ exists and $M$ is essentially constant with value $X$.

**Proof.**
Omitted. This is a good excercise in the definitions.
$\square$

Example 4.22.6. Let $\mathcal{C}$ be a category. Let $(X_ n)$ and $(Y_ n)$ be inverse systems in $\mathcal{C}$ over $\mathbf{N}$ with the usual ordering. Picture:

\[ \ldots \to X_3 \to X_2 \to X_1 \quad \text{and}\quad \ldots \to Y_3 \to Y_2 \to Y_1 \]

Let $a : (X_ n) \to (Y_ n)$ be a morphism of pro-objects of $\mathcal{C}$. What does $a$ amount to? Well, for each $n \in \mathbf{N}$ there should exist an $m(n)$ and a morphism $a_ n : X_{m(n)} \to Y_ n$. These morphisms ought to agree in the following sense: for all $n' \geq n$ there exists an $m(n', n) \geq m(n'), m(n)$ such that the diagram

\[ \xymatrix{ X_{m(n, n')} \ar[rr] \ar[d] & & X_{m(n)} \ar[d]^{a_ n} \\ X_{m(n')} \ar[r]^{a_{n'}} & Y_{n'} \ar[r] & Y_ n } \]

commutes. After replacing $m(n)$ by $\max _{k, l \leq n}\{ m(n, k), m(k, l)\} $ we see that we obtain $\ldots \geq m(3) \geq m(2) \geq m(1)$ and a commutative diagram

\[ \xymatrix{ \ldots \ar[r] & X_{m(3)} \ar[d]^{a_3} \ar[r] & X_{m(2)} \ar[d]^{a_2} \ar[r] & X_{m(1)} \ar[d]^{a_1} \\ \ldots \ar[r] & Y_3 \ar[r] & Y_2 \ar[r] & Y_1 } \]

Given an increasing map $m' : \mathbf{N} \to \mathbf{N}$ with $m' \geq m$ and setting $a'_ i : X_{m'(i)} \to X_{m(i)} \to Y_ i$ the pair $(m', a')$ defines the same morphism of pro-systems. Conversely, given two pairs $(m_1, a_1)$ and $(m_1, a_2)$ as above then these define the same morphism of pro-objects if and only if we can find $m' \geq m_1, m_2$ such that $a'_1 = a'_2$.

Lemma 4.22.8. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered (resp. cofiltered) index category $\mathcal{I}$. Let $F : \mathcal{C} \to \mathcal{D}$ be a functor. If $M$ is essentially constant as an ind-object (resp. pro-object), then so is $F \circ M : \mathcal{I} \to \mathcal{D}$.

**Proof.**
If $X$ is a value for $M$, then it follows immediately from the definition that $F(X)$ is a value for $F \circ M$.
$\square$

Lemma 4.22.9. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered index category $\mathcal{I}$. The following are equivalent

$M$ is an essentially constant ind-object, and

$X = \mathop{\mathrm{colim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \]

is bijective.

**Proof.**
Assume (2) holds. Then $\text{id}_ X \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, X)$ comes from a morphism $X \to M_ i$ for some $i$, i.e., $X \to M_ i \to X$ is the identity. Then both maps

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \]

are bijective for all $W$ where the first one is induced by the morphism $X \to M_ i$ we found above, and the composition is the identity. This means that the composition

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \]

is the identity too. Setting $W = M_ j$ and starting with $\text{id}_{M_ j}$ in the colimit, we see that $M_ j \to X \to M_ i \to M_ k$ is equal to $M_ j \to M_ k$ for some $k$ large enough. This proves (1) holds. The proof of (1) $\Rightarrow $ (2) is omitted.
$\square$

Lemma 4.22.10. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with cofiltered index category $\mathcal{I}$. The following are equivalent

$M$ is an essentially constant pro-object, and

$X = \mathop{\mathrm{lim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map

\[ \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) \]

is bijective.

**Proof.**
Assume (2) holds. Then $\text{id}_ X \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, X)$ comes from a morphism $M_ i \to X$ for some $i$, i.e., $X \to M_ i \to X$ is the identity. Then both maps

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) \]

are bijective for all $W$ where the first one is induced by the morphism $M_ i \to X$ we found above, and the composition is the identity. This means that the composition

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \]

is the identity too. Setting $W = M_ j$ and starting with $\text{id}_{M_ j}$ in the colimit, we see that $M_ k \to M_ i \to X \to M_ j$ is equal to $M_ k \to M_ j$ for some $k$ large enough. This proves (1) holds. The proof of (1) $\Rightarrow $ (2) is omitted.
$\square$

Lemma 4.22.11. Let $\mathcal{C}$ be a category. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of filtered index categories. If $H$ is cofinal, then any diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ H$ is essentially constant.

**Proof.**
This follows formally from Lemmas 4.22.9 and 4.17.2.
$\square$

Lemma 4.22.12. Let $\mathcal{I}$ and $\mathcal{J}$ be filtered categories and denote $p : \mathcal{I} \times \mathcal{J} \to \mathcal{J}$ the projection. Then $\mathcal{I} \times \mathcal{J}$ is filtered and a diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ p : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ is essentially constant.

**Proof.**
We omit the verification that $\mathcal{I} \times \mathcal{J}$ is filtered. The equivalence follows from Lemma 4.22.11 because $p$ is cofinal (verification omitted).
$\square$

Lemma 4.22.13. Let $\mathcal{C}$ be a category. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of cofiltered index categories. If $H$ is initial, then any diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ H$ is essentially constant.

**Proof.**
This follows formally from Lemmas 4.22.10, 4.17.4, 4.17.2, and the fact that if $\mathcal{I}$ is initial in $\mathcal{J}$, then $\mathcal{I}^{opp}$ is cofinal in $\mathcal{J}^{opp}$.
$\square$

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