## 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 exists 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 $\mathcal{C}$.

1. 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 exists an $i \in \mathcal{I}$ and a morphism $X \to M_ i$ such that

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

2. 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 $M_ j \to X \to M_ i \to M_ k$.

2. 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 exists an $i \in \mathcal{I}$ and a morphism $M_ i \to X$ such that

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

2. 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. In general this need not be the case however. An example is 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 $\mathbf{Z}$. A non-example is to 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. $\square$

Remark 4.22.4. Let $\mathcal{C}$ be a category. There exists a big category $\text{Ind-}\mathcal{C}$ of ind-objects of $\mathcal{C}$. Namely, if $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ are filtered diagrams in $\mathcal{C}$, then we can define

$\mathop{Mor}\nolimits _{\text{Ind-}\mathcal{C}}(F, G) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ j \mathop{Mor}\nolimits _\mathcal {C}(F(i), G(j)).$

There is a canonical functor $\mathcal{C} \to \text{Ind-}\mathcal{C}$ which maps $X$ to the constant system on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here.

Remark 4.22.5. Let $\mathcal{C}$ be a category. There exists a big category $\text{Pro-}\mathcal{C}$ of pro-objects of $\mathcal{C}$. Namely, if $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ are cofiltered diagrams in $\mathcal{C}$, then we can define

$\mathop{Mor}\nolimits _{\text{Pro-}\mathcal{C}}(F, G) = \mathop{\mathrm{lim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _\mathcal {C}(F(i), G(j)).$

There is a canonical functor $\mathcal{C} \to \text{Pro-}\mathcal{C}$ which maps $X$ to the constant system on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here.

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 $a'_1 = a'_2$.

Remark 4.22.7. Let $\mathcal{C}$ be a category. Let $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ be cofiltered diagrams in $\mathcal{C}$. Consider the functors $A, B : \mathcal{C} \to \textit{Sets}$ defined by

$A(X) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _\mathcal {C}(F(i), X) \quad \text{and}\quad B(X) = \mathop{\mathrm{colim}}\nolimits _ j \mathop{Mor}\nolimits _\mathcal {C}(G(j), X)$

We claim that a morphism of pro-systems from $F$ to $G$ is the same thing as a transformation of functors $t : B \to A$. Namely, given $t$ we can apply $t$ to the class of $\text{id}_{G(j)}$ in $B(G(j))$ to get a compatible system of elements $\xi _ j \in A(G(j)) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _\mathcal {C}(F(i), G(j))$ which is exactly our definition of a morphism in $\text{Pro-}\mathcal{C}$ in Remark 4.22.5. We omit the construction of a transformation $B \to A$ given a morphism of pro-objects from $F$ to $G$.

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

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

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

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

is bijective.

Proof. Assume (2) holds. Then $\text{id}_ X \in \mathop{Mor}\nolimits _\mathcal {C}(X, X)$ 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

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

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

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

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

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

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

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

is bijective.

Proof. Assume (2) holds. Then $\text{id}_ X \in \mathop{Mor}\nolimits _\mathcal {C}(X, X)$ 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

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

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

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

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 $\mathcal{J}^{opp}$. $\square$

Comment #538 by Nuno on

Minor typos: (i) "Then $X$ comes equipped with projection morphisms $M_i \to X$." I believe projection should be injection here. (ii) "essentially constant as an pro-object".

Comment #4831 by Weixiao Lu on

I think it should be mentioned in Definition 05PU that $X = \varinjlim_{i\in\mathcal I}M_i$ can be derived from conditions (a) and (b).

Comment #4837 by on

@#4831. Parts (a) and (b) do not make sense if you do not already know that $X = \lim M_i$ because we are using the coprojections $M_i \to X$ to formulate them. Moreover, I wanted to stress that given an essentially constant system, we have a colimit of the system with expected value.

Comment #5544 by on

But it is enough to suppose that $X$ is a (co-)cone of the diagram, and this observation is actually important in Lemma 05SH, as this lemma is (rightly) formulated without any exactness conditions on $F$. Such conditions are not needed because any functor preserves (co-)cones even if it does not preserve (co-)limits.

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