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 exercise 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,
there exists a cocone (X, \{ M_ i \to X\} _ i) such that for any W in \mathcal{C} the map \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) is bijective,
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) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) is bijective, and
there exists an i in \mathcal{I} and a morphism X \to M_ i such that for any W in \mathcal{C} the map \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \to \mathop{\mathrm{colim}}\nolimits _{j \geq i} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ j) is bijective.
In cases (2), (3), and (4) the value of the essentially constant system is X.
Proof.
It is clear that (3) implies (2). Assume (2). 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.
Assume (4). Let k be an object of \mathcal{I}. Setting W = M_ k we deduce there exists a unique morphism M_ k \to X such that there exists a j and morphisms k \to j and i \to j in \mathcal{I} such that M_ k \to X \to M_ i \to M_ j is equal to M_ k \to M_ j. The uniqueness guarantees that we obtain a cocone (X, \{ M_ k \to X\} ). In this way we see that (4) implies (2); some details omitted.
We omit the proof that (1) implies the other conditions.
\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,
there exists a cone (X, \{ X \to M_ i\} ) such that 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) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) is bijective,
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) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) is bijective, and
there exists an i in \mathcal{I} and a morphism M_ i \to X such that for any W in \mathcal{C} the map \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) \to \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) is bijective.
In cases (2), (3), and (4) the value of the essentially constant system is X.
Proof.
This lemma is dual to Lemma 4.22.9.
\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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