We now consider the condensed analogues of abelian groups. This will implicitly tell us things about condensed \(R\)-modules for any (discrete) commutative ring \(R\text{,}\) as these will be defined as \(R\)-module objects in condensed abelian groups.
Reference.
This lecture is based on [8], Lectures 2 and 3. Proposition 5.4.2 is taken from [8], Lecture 7 (timestamp 5:00).
Subsection5.1Condensed abelian groups
Since we are now talking about sheaves on a Grothendieck topology, we must distinguish between sheaves and presheaves.
Definition5.1.1.
The forgetful functor from \(\CSet\) to \(\Fun(\Prof^{\op}, \Set)\) (i.e., presheaves in sets on \(\Prof\)) has a left adjoint called sheafification. Concretely, given \(F \in \Fun(\Prof^{\op}, \Set)\text{,}\) its sheafification evaluates on \(S\) to the colimit of \(\prod_i F(T_i)\) over all finite jointly surjective families of morphisms \(T_i \to S\) in \(\Prof\text{.}\) (Note that this makes sense because \(\Prof\) is a small category.)
Definition5.1.2.
We define the category \(\CAb\) of condensed abelian groups as the category of abelian group objects in \(\CSet\text{.}\) That is, an object of \(\CAb\) is a functor in \(\Fun(\Prof^{\op}, \Ab)\) which when composed to \(\Ab \to \Set\) yields an object of \(\CSet\) (meaning that it satisfies the same compatibilities with finite disjoint unions and arbitrary surjections).
The category \(\CAb\) is abelian, but the cokernels are obtained by taking the presheaf cokernel (i.e., point by point) and then sheafifying.
The category \(\CAb\) contains the sequentially topologized abelian groups \(\TopAb^{\seq}\) as a full subcategory. In particular it contains the discrete abelian groups \(\Ab\) as a full subcategory.
Remark5.1.3.
Beware that the functor \(\CSet \to \Top\) (Definition 4.3.1) does not promote to a functor \(\CAb \to \TopAb\text{!}\) The problem (as noted in Remark 4.3.3) is that this functor does not preserve products, so for instance the multiplication map \(X \times X \to X\) of an object of \(\CSet\) does not transfer to a corresponding map of topological spaces. This is one reason why the theory we are about to described cannot be mirrored in topological abelian groups; another is the existence of examples of bad topological quotients which nonetheless work well in condensed abelian groups, such as Example 5.1.4.
Example5.1.4.
There is a natural homomorphism \(\RR_{\disc} \to \RR\) of topological abelian groups. Applying the functor \(\Top \to \CSet\text{,}\) we obtain a homomorphism in \(\CAb\text{,}\) which therefore has a cokernel \(\underline{\RR}/\underline{\RR}_{\disc}\text{.}\) Explicitly, for \(S \in \Prof\text{,}\)\(\underline{\RR}(S)\) consists of continuous maps \(S \to \RR\) while \(\underline{\RR}_{\disc}(S)\) consists of locally constant functions \(S \to \RR\text{.}\) In this case, we showed in Example 4.1.9 that there is no sheafification required: \((\underline{\RR}/\underline{\RR}_{\disc})(S) = \underline{\RR}(S)/\underline{\RR}_{\disc}(S)\text{.}\) See Remark 7.3.8 for another explanation of this observation.
Definition5.1.5.
There is an obvious forgetful functor \(\CAb \to \CSet\text{.}\) This has a left adjoint \(X \mapsto \ZZ[X]\) where \(\ZZ[X]\) denotes the sheafification of the sheaf \(S \mapsto \ZZ[X(S)]\) (and \(\ZZ[X(S)]\) denotes the free \(\ZZ\)-module on the set \(X(S)\)). The sheafification is already needed for trivial reasons: the empty set needs to go to 0 rather than to \(\ZZ[X(\emptyset)] \cong \ZZ\text{.}\)
As an aside, it would be a bit more consistent to denote \(\ZZ[X]\) by \(\underline{\ZZ}[X]\text{,}\) but we will not bother doing this.
Example5.1.6.
For \(S \in \Set\) corresponding to \(\underline{S} \in \CSet\text{,}\)\(\ZZ[\underline{S}]\) represents the discrete abelian group \(\ZZ[S]\text{.}\)
Example5.1.7.
For \(X \in \Prof\text{,}\) we claim that \(\ZZ[\underline{X}]\) represents a locally compact topological group \(G\) which can be described as follows. For each \(n \in \NN\text{,}\) let \(G_n\) be the quotient of \(X^{n}\) by the action of the symmetric group \(S_n\text{.}\) We then get a monoid structure on \(G_+ := \sqcup_{n \in \NN} G_n\) induced by the maps \(X^{m} \times X^n \cong X^{m+n}\text{.}\) Finally, let \(G\) be the group completion of \(G_+\text{,}\) i.e., the quotient of \(G_+ \times G_+\) by the relation that \((a,b) \sim (c,d)\) iff \(a+d = b+c\text{;}\) this is the colimit of the compact subspaces \(G_n - G_n = \{x-y\colon x,y \in G_n\}\) for \(n \in \NN\text{.}\)
To see that this description is correct, note first that the presheaf \(S \mapsto \ZZ[X(S)]\) maps injectively to \(\underline{G}\) by mapping each element of \(X(S)\) to the corresponding map from \(S\) to \(G_1 \cong X\text{.}\) This induces an injective map \(\ZZ[\underline{X}] \to \underline{G}\text{.}\) To check that this map is surjective, note that any continuous map \(S \to G\) factors through \(G_n - G_n\) for some \(n\text{.}\) We can view \(G_n\) as a quotient of \(X^{2n}\text{;}\) then for some covering \(S' \to S\) the composition \(S' \to S \to G_n\) factors through \(X^{2n}\) (e.g., take \(S' = S \times_{G_n-G_n} X^{2n}\)). We can then lift the element of \(\underline{G}(S)\) corresponding to the chosen map to an element of \(\ZZ[\underline{X}](S')\text{,}\) proving the claim.
Using Remark 4.2.2, one can upgrade the previous discussion to cover \(X \in \CHaus\text{.}\) By taking colimits, we can make a similar description for \(X\) a locally compact topological space (where now the colimit in the definition of \(G\) runs over both \(n\) and compact subspaces of \(X\)).
Remark5.1.8.
In Example 5.1.7, the map \(X_{\disc} \to X\) induces an injection \(\ZZ[X] = \ZZ[\underline{X_{\disc}}] \to \ZZ[\underline{X}]\) which is a bijection of underlying groups but not a homeomorphism. The maps \(X \to X_i\) induce a natural map \(\ZZ[\underline{X}] \to \varprojlim_i \ZZ[X_i]\) at the level of topological abelian groups, but this map is not a bijection even on underlying groups! One way to see this is to interpret \(\ZZ[X]\) as the set of \(\ZZ\)-linear combinations of Dirac measures on \(X\text{,}\) and \(\varprojlim_i \ZZ[X_i]\) as the full set of \(\ZZ\)-valued measures on \(X\text{;}\) the latter will end up being the solidification of \(\ZZ[\underline{X}]\) (Lemma 7.3.7).
Example5.1.9.
Any \(X \in \Set\) is projective as an object of \(\CSet\text{;}\) by adjunction, \(\ZZ[\underline{X}]\) is projective in \(\CAb\text{.}\) In particular, for any singleton \(* \in \Prof\text{,}\) the functor \(\CAb \to \Ab\) given by evaluation at \(*\) is exact.
By contrast, for \(X \in \Prof\text{,}\) the objects \(\ZZ[\underline{X}]\) in \(\CAb\) form a collection of compact generators, but they are in general not projective. Namely, projectivity would say that for \(Z \to Y\) an epimorphism in \(\CAb\text{,}\) the map \(Z(X) \to Z(Y)\) is again surjective; this is not guaranteed because objects of \(\CAb\) are sheaves rather than presheaves (Remark 4.1.2; compare also Remark 3.3.5). In fact, projective objects in \(\CAb\) are somehat hard to come by; see Proposition 5.4.8 for a key exception.
Subsection5.2Internal Hom and tensor product
Definition5.2.1.
We obtain a symmetric monoidal tensor product \(\otimes\) with unit \(\ZZ[*]\) by sheafifying the pointwise tensor product. For \(X,Y \in \CSet\text{,}\) we have a natural identification
Although it may look innocuous, Definition 5.2.1 is a crucial point in the theory! Despite the fact that the basic objects \(\ZZ[\underline{X}]\) for \(X \in \Prof\) are all represented by topological (even locally compact) abelian groups, and \(\otimes\) acts on these objects via (5.1), the flexibility to sheafify over coverings of and by profinite sets will prove to be indispensable.
Example5.2.3.
For \(S \in \Prof\text{,}\)\(X \in \TopAb\text{,}\) we compute \(\iHom_{\CAb}(\ZZ[\underline{S}], \underline{X})\) as a functor: for \(T \in \Prof\text{,}\)
The same logic applies when \(S \in \Top^{\seq}\) is locally compact: by Proposition 4.3.2 and Proposition 4.4.2, we still have \(\Hom_{\CSet}(\underline{S \times T}, \underline{X}) = \Cts(S \times T, X)\text{,}\) and when \(S\) is locally compact we again have \(\Cts(S \times T, X) = \Cts(T, \Cts(S, X))\text{.}\)
Remark5.2.4.
For any \(X \in \CSet\text{,}\) tensoring with \(\ZZ[X]\) has the effect of tensoring with \(S \mapsto \ZZ[X(S)]\) at the presheaf level and then sheafifying. As a result, this operation is exact; that is, \(\ZZ[X]\) is flat.
Definition5.2.5.
By the general argument of Grothendieck, \(\CAb\) contains enough injective objects. Since the functor \(\Hom_{\CAb}(M, \bullet)\) is left exact, we may define right derived functors \(\Ext^i_{\CAb}(M, \bullet)\) and even the full derived functor on the bounded-below derived category. In light of (5.2), we define the internal Ext object \(\iExt^i_{\CAb}(M, N)\) by sheafifying the presheaf \(S \mapsto \Ext^i_{\CAb}(M \otimes \ZZ[\underline{S}], N)\text{.}\)
Subsection5.3The sequence space
The definition of a condensed abelian group makes it quite hard to work with any examples. We next introduce an object that will be key to a further elucidation of the category.
Definition5.3.1.
Define the sequence space as the object \(P := \ZZ[\underline{\NN_\infty}]/\ZZ[\underline{\infty}]\) in \(\CAb\text{.}\) We may promote \(P\) to a ring object in \(\CAb\) using the fact that the addition map \(\NN \times \NN \to \NN\) extends to a continuous map \(\NN_\infty \times \NN_\infty \to \NN_\infty\) by the rule that \(\bullet + \infty = \infty + \bullet = \infty\text{.}\) The composition \(\ZZ[\NN] \to \ZZ[\underline{\NN_\infty}]\) then becomes a ring structure when we identify \(\ZZ[\NN]\) with \(\ZZ[x]\) via \([n] \mapsto x^n\text{.}\)
Since the inclusion \(\ZZ[\underline{\infty}] \to \ZZ[\underline{\NN_\infty}]\) is split by the map corresponding to the unique projection \(\NN_\infty \to \infty\text{,}\) we have an isomorphism \(\ZZ[\underline{\NN_\infty}] \cong \ZZ[\underline{\infty}] \oplus P\) in \(\CAb\text{.}\) For a generalization of this splitting construction, see Lemma 5.4.1.
We see from this that \(P \in \CAb\) is represented by a topological abelian group with underlying group \(\ZZ[\NN]\) equipped with the subspace topology from \(\ZZ[\underline{\NN_\infty}]\) (Example 5.1.7). For example, the sequence \([0], [1], \dots\) is a null sequence for this topology.
Remark5.3.2.
Recall from Remark 4.1.7 that for \(X\) a Hausdorff topological group, \(\Hom_{\CAb}(\ZZ[\underline{\NN_\infty}], \underline{X})\) can be interpreted as the space of convergent sequences in \(X\text{.}\) By the same token, \(\Hom_{\CAb}(P, \underline{X})\) can be interpreted as the space of null sequences in \(X\text{;}\) for \(n \in \NN\text{,}\) precomposing with the map \(\underline{\ZZ} \to P\) corresponding to \(\{n\} \to \NN_\infty\) gives the \(n\)-th coordinate map \(\Hom_{\CAb}(P, \underline{X}) \to \Hom_{\CAb}(\underline{\ZZ}, \underline{X}) = X\) on null sequences.
In particular this applies when \(X \in \Prof\text{,}\) but note that this is the “wrong way around”: an object of \(\CSet\) is characterized by its maps out of profinite sets, not into them. Nonetheless it may be helpful to keep this interpretation of \(P\) in mind going forward, in order to be able to parse the following arguments.
We highlight one concrete example. For \(n \in \NN\text{,}\) the sequence valued in \(\ZZ\) with a 1 in position \(n\) and 0 elsewhere is a null sequence for the discrete topology; it thus corresponds to a map \(c_n\colon P \to \underline{\ZZ}\) taking \([n]\) to 1 and \([m]\) to 0 for \(m \neq n\text{.}\) Another way to obtain this map is to apply adjunction to the morphism \(\NN_\infty \to \{n\}\) in \(\Prof\) and then factor the resulting map \(\ZZ[\underline{\NN_\infty}] \to \ZZ[\underline{\{n\}}] \cong \ZZ\) through \(P\text{.}\) Putting the maps \(c_n\) together yields the coordinate map \(c\colon P \to \prod_\NN \underline{\ZZ}\) which has dense image; this statement will later be upgraded to say this map induces an isomorphism of solidifications (Lemma 7.1.2).
Subsection5.4Projectivity of the sequence space
We will study \(P\) using the following general observations. Together these generalize the fact that for \(X_1,X_2 \in \Prof\) and \(X := X_1 \sqcup X_2\text{,}\) we have a canonical isomorphism \(\ZZ[\underline{X}] \cong \ZZ[\underline{X_1}] \oplus \ZZ[\underline{X_2}]\text{.}\)
Lemma5.4.1.
Choose \(X \in \Prof\text{,}\) let \(Z \subseteq X\) be a closed subspace, and set \(U := X \setminus Z\text{.}\) Then \(\ZZ[\underline{X}]/\ZZ[\underline{Z}]\) is a direct summand of \(\ZZ[\underline{X}]\) in \(\CAb\text{.}\)
Proof.
Since \(Z\) is injective in \(\Prof\) (Proposition 3.3.3), the injection \(Z \to X\) admits a retraction \(X \to Z\text{.}\) Now consider the map \(X \to \ZZ[\underline{X}]\) given by the difference between the canonical map and the composition \(X \to Z \to X \to \ZZ[\underline{X}]\text{.}\) As the two maps agree on \(Z\text{,}\) the induced map \(\ZZ[\underline{X}] \to \ZZ[\underline{X}]\) factors through a map \(\ZZ[\underline{X}]/\ZZ[\underline{Z}] \to \ZZ[\underline{X}]\text{,}\) yielding the claimed splitting.
Proposition5.4.2.
With notation as in Lemma 5.4.1, choose also \(X' \in \Prof\text{,}\) let \(Z' \subseteq X\) be a closed subspace, and set \(U' := X' \setminus Z'\text{.}\) Then any homeomorphism \(U \cong U'\) induces an isomorphism \(\ZZ[\underline{X}]/\ZZ[\underline{Z}] \cong \ZZ[\underline{X'}]/\ZZ[\underline{Z'}]\) compatible with the isomorphism of discrete groups \(\ZZ[U] \cong \ZZ[U']\text{.}\)
Proof.
We may formally reduce to the case where \(Z = \{\infty\}\) is a singleton (by comparing other two cases to this one). Then the homeomorphism \(U' \cong U\) extends to a surjection \(X' \to X\) in \(\Prof\text{;}\) we thus have a surjection \(\ZZ[\underline{X'}]/\ZZ[\underline{Z'}] \to \ZZ[\underline{X}]/\ZZ[\underline{Z}]\) which it will suffice to split.
Apply Lemma 5.4.1 to construct a morphism \(\ZZ[\underline{X}']/\ZZ[\underline{Z}'] \to \ZZ[\underline{X'}]\) in \(\CAb\) splitting the canonical projection. Since the composition \(X' \to \ZZ[\underline{X}']/\ZZ[\underline{Z}'] \to \ZZ[\underline{X'}]\) maps all of \(Z'\) to a single element, it factors through \(X\text{.}\) This gives us a map \(\ZZ[\underline{X}] \to \ZZ[\underline{X}']/\ZZ[\underline{Z}']\) inducing the desired splitting.
Corollary5.4.3.
With notation as in Lemma 5.4.1, the assignment \(U \mapsto \ZZ[\underline{X}]/\ZZ[\underline{Z}]\) is functorial in \(U\text{,}\) as an object of the full subcategory of \(\Top\) consisting of spaces which occur as open subsets of spaces in \(\Prof\text{.}\)
Proof.
It suffices to observe that with notation as in Proposition 5.4.2, if we are given a morphism \(U' \to U\) in \(\Top\text{,}\) we can extend it to a morphism \(X'' \to X\) where \(X''\) is the closure of the image of the map \(U' \to X' \times X\text{.}\) Replacing \(X'\) with \(X''\) has no effect per Proposition 5.4.2.
Remark5.4.4.
One application of Proposition 5.4.2 is to show that \(P\) is projective in \(\CAb\text{,}\) as follows. For any epimorphism \(N' \to N\) in \(\CAb\) and any map \(P \to N\text{,}\) we can lift the underlying map \(\NN_\infty \to N\) to a map \(S \to N'\) for some surjection \(S \to N_\infty\) in \(\Prof\text{.}\) Let \(S_\infty\) be the fiber of \(S\) above \(\infty\text{;}\) we obtain a diagram of the form
Figure5.4.5. by precomposing the induced map \(\ZZ[\underline{S}] \to N'\) with a splitting \(\ZZ[\underline{S}]/\ZZ[\underline{S_\infty}] \to \ZZ[\underline{S}]\) from Lemma 5.4.1.
By picking one element of \(S\) above each \(n \in \NN\) but retaining the entire fiber \(S_\infty\) above \(\infty\text{,}\) we may find another compactification \(S'\) of \(\NN_\infty\) admitting a map \(S' \to S\) such that \(S' \to S \to \NN_\infty\) restricts to an isomorphism over \(\NN\text{.}\) After pulling back the previous diagram to obtain
Figure5.4.6. we may apply Proposition 5.4.2 to see that the left column is an isomorphism; we thus get a map \(P \to N'\) lifting \(P \to N\text{.}\)
Note that the projectivity of \(P\) is really a phenomenon of condensed abelian groups; it does not come from a corresponding statement at the level of condensed sets because \(\underline{\NN_\infty}\) is not projective even in \(\Prof\) (Remark 3.3.5), let alone in \(\CSet\text{.}\)
We next give a strong generalization of Remark 5.4.4.
Definition5.4.7.
An object \(M \in \CAb\) is internally projective if the functor \(\iHom_{\CAb}(M, \bullet)\) on \(\CAb\) is exact. In particular, any such object is projective.
For example, for any singleton set \(*\text{,}\)\(\ZZ[\underline{*}]\) is internally projective because the functor \(\iHom_{\CAb}(\ZZ[\underline{*}], \bullet)\) is the identity.
Proposition5.4.8.
The object \(P \in \CAb\) is internally projective. By Definition 5.3.1, the same is then true of \(\ZZ[\underline{\NN_\infty}]\text{.}\)
Proof.
It will suffice to check a sheafified version of the usual arrow-theoretic definition of projectivity: for any \(T \in \Prof\text{,}\) given an epimorphism \(N' \to N\) in \(\CAb\) and a morphism \(P \otimes \ZZ[\underline{T}] \to N\text{,}\) we can pull back the latter along some surjection \(T' \to T\) in \(\Prof\) in such a way that the resulting morphism \(P \otimes \ZZ[\underline{T}'] \to N\) can be factored through \(N'\text{.}\) (Note that we do have to sheafify; we cannot hope to prove that \(P \otimes \ZZ[\underline{T}]\) is itself projective in \(\CAb\text{.}\))
We proceed by adapting Remark 5.4.4 to this setting. We again have a surjection \(S \to \NN_\infty \times T\) in \(\Prof\) such that the element of \(N(\NN_\infty \times T) = \Hom_{\CSet}(\underline{\NN_\infty \times T}, N)\) corresponding to \(\ZZ[\underline{\NN_\infty \times T}] \to P \otimes \ZZ[\underline{T}] \to N\) via adjunction lifts to \(N'(S)\text{.}\) For \(n \in \NN_\infty\text{,}\) let \(S_n\) denote the fiber of \(S \to \NN_\infty\) over \(n\text{;}\) we again obtain a diagram of the form
Figure5.4.9. by applying Lemma 5.4.1 to the pair \((S, S_\infty)\text{.}\)
We next choose a covering \(T' \to T\) with the property that for each \(n \in \NN\text{,}\) the induced covering \(S_n \times_{T} T' \to T'\) splits. For instance, we could take \(T' = T \times \prod_{n \in \NN} S_n\) using the first projection to \(T\text{;}\) we get a splitting \(T' \to S_n \times_T T'\) by taking the projection \(T' \to S_n\) in the first factor and the identity in the second factor.
Now let \(S'\) be the closed subset of \(S\) consisting of \(S_\infty\) together with the image of one splitting of \(S_n \to T'\) for each \(n \in \NN\text{;}\) then \(S' \to S \to \NN_\infty \times T'\) is a surjection which restricts to a homeomorphism over \(\NN \times T'\text{.}\) We may thus pull back the previous diagram along Proposition 5.4.2 to identify
In principle, the proof of Proposition 5.4.8 also shows that if \(X \in \Prof\text{,}\)\(Z \subseteq X\) is closed, and the subspace topology on \(U := X \setminus Z\) is discrete, then \(\ZZ[\underline{X}]/\ZZ[\underline{Z}]\) is internally projective in \(\CAb\text{.}\) However, in this case \(X \setminus Z\) is either finite or countable, so this just reduces to saying that \(\ZZ\) and \(P\) are internally projective.
