Continuous Relative Completion

• Continuous relative completion is a categorical framework that unifies various completion processes (e.g., Cauchy, sheafification, group completions) using enriched monads and proper factorization systems.
• It provides functorial and universal completion methods that preserve cohomological, finiteness, and representation-theoretic properties across algebraic, topological, and analytical contexts.
• Applications include Schlichting completion in tdlc groups, Tannakian completions in arithmetic geometry, and explicit models in algebraic de Rham theory and modular forms.

Continuous relative completion unifies several abstract and concrete processes—such as Cauchy completion in normed spaces, sheafification in topos theory, and relative (pro-)algebraic completions of discrete or topological groups—under a categorical and homological framework. It provides a functorial method to “complete” objects relative to a monad or a subgroup/topological structure, preserving or controlling various finiteness, cohomological, or representation-theoretic properties in a continuous or enriched sense. Major paradigms include the enriched monad-theoretic approach, topological group completions (Schlichting completion), Tannakian relative completions in arithmetic geometry, and their explicit algebraic and Hodge/de Rham incarnations.

1. Abstract Foundations and Monad-Theoretic Framework

Continuous relative completion in the categorical framework is formalized in terms of enriched categories, proper factorization systems, and monads. Let $\mathcal{V}$ be a symmetric monoidal closed category, $\mathcal{B}$ a $\mathcal{V}$-category equipped with a proper $\mathcal{V}$-prefactorization system $(\mathcal{E},\mathcal{M})$, and $\mathbb{T} = (T,\eta,\mu)$ a $\mathcal{V}$-monad on $\mathcal{B}$. One introduces:

• $T$-embeddings: morphisms in $\mathcal{M}$.
• $T$-closed embeddings: $m \in \mathcal{M}$ such that for all $f \in \Sigma_T := \{ f \mid Tf\ \text{is iso} \}$, $f \downarrow_{\mathcal{V}} m$.
• $T$-dense morphisms: $f \in \mathcal{E}$ such that for all $m$ as above, $f \downarrow_{\mathcal{V}} m$.

Assuming every morphism factors as a $T$-dense map followed by a $T$-closed $\mathcal{M}$-embedding, one acquires an enriched factorization system $$(\Sigma_T^{\uparrow_\mathcal{V}},\Sigma_T^{\downarrow_\mathcal{V}})$$, leading to an idempotent closure operator. For each $\mathcal{M}$-subobject $m:X'\hookrightarrow X$, the $T$-closure $\overline{m}: \overline{X} \hookrightarrow X$ is determined as the $\mathcal{M}$-factor in this system (Lucyshyn-Wright, 2014).

2. Idempotent Core and Universal Properties

The idempotent core of a $\mathcal{V}$-monad $T$ is the terminal idempotent $\mathcal{V}$-monad $\widetilde{T}$ inverting the same morphisms as $T$. Explicitly, the full subcategory

$\mathcal{B}_{(T)} := \{ B \in \mathcal{B} \mid B\ \text{is $T$-complete and$T$-separated} \}$

is reflective. The universal property: for any idempotent $\mathcal{V}$-monad $S$ with a monad morphism $S \to T$, there exists a unique factorization $S \to \widetilde{T} \to T$, making $\widetilde{T}$ the terminal idempotent monad over $T$ (Lucyshyn-Wright, 2014). The practical import is that $T$-completion can be functorially and universally constructed given suitable factorization, subsuming classical and novel completion phenomena.

3. Schlichting Completion and Finiteness Transfer

In the context of totally disconnected locally compact (tdlc) groups, the Schlichting (continuous relative) completion of a discrete group $G$ relative to a commensurated subgroup $H$ is defined as follows. $G$ acts on the coset set $G/H$ with closure $\hat{G} := G//H$ inside $\mathrm{Sym}(G/H)$ (pointwise convergence topology). The image of $H$ is dense in a compact–open subgroup $U = \overline{\alpha(H)}$ of $\hat{G}$, and $G$ embeds densely in $\hat{G}$ if $\cap_{g\in G}gHg^{-1}=1$.

This construction yields the following key transfer results (Bonn et al., 2024):

• If $G$ and $H$ are of type $\mathrm{FP}_n^R$ (e.g., type $F_n$), so is the Schlichting completion $\hat{G}$.
• Cohomological isomorphisms: for $H$ locally finite, the restriction $H^*_{{\rm cont}}(\hat{G},M)\to H^*(G,M)$ is an isomorphism for all $R$-modules $M$ with trivial $\hat{G}$-action.
• Applications include vanishing theorems for Neretin groups and controlled construction of tdlc groups with prescribed finiteness/cohomological properties.

The bridge between discrete and continuous invariants is realized by permutation resolutions and averaging over compact open subgroups.

4. Tannakian and Motivic Approaches to Relative Completion

Relative completion admits a Tannakian description: for a discrete group $\Gamma$ and Zariski-dense homomorphism $\rho:\Gamma\to R(K)$ to a reductive $K$-group $R$, the continuous relative completion is defined as the pro-algebraic group $$\widehat{\Gamma}_R := \mathrm{Aut}^\otimes(\omega|_{\mathrm{Rep}(\Gamma)^{\mathrm{rel}}})$$ over $K$ (Kantor, 2020). This group fits into an exact sequence

$1 \to U \to \widehat{\Gamma}_R \to R \to 1$

with $U$ pro-unipotent. The presentation of $\operatorname{Lie} U$ arises from completed free Lie algebras on $H^1(\Gamma,\mathcal{O}_R)^*$ modulo relations from $H^2(\Gamma,\mathcal{O}_R)^*$.

Comparison isomorphisms exist between Betti and de Rham realizations; the theory underlies non-abelian Chabauty-Kim methods, Selmer stacks, and period maps in diophantine geometry (Kantor, 2020).

5. Explicit Models: Algebraic de Rham, Modular Forms, and Multiple Modular Values

For $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, the relative completion over $\mathbb{Q}$ gives a pro-algebraic group $G_{\rm rel}$ with unipotent radical generated by cohomology $$H^1(\mathrm{SL}_2(\mathbb{Z}), S^{2n}H)^* \otimes S^{2n}H$$ ($H$ the standard 2-dimensional $\mathrm{SL}_2$-representation) (Luo, 2018). This structure is encoded in an explicit Hopf algebra model, whose coordinate ring features:

• Duals of group cohomology in the Lie algebra and completed commutative Hopf algebra structure.
• Filtrations (weight, Hodge) stemming from mixed Hodge theory (e.g., Eichler-Shimura).
• Universal Gauss–Manin connection $\nabla^{dR} = \nabla_{GM} + \Omega$ on the unipotent piece, with $\Omega$ constructed from modular forms and modular forms of the second kind.

Iterated integrals associated with these structures generalize classical Eichler integrals, yielding all multiple modular values as periods of the pro-unipotent fundamental group.

6. Concreteness: Functional Analysis and Sheafification Examples

The abstract formalism covers classical completions:

• Normed vector spaces: For $B$ the category of normed spaces, $(\mathcal{E}$ = surjections, $\mathcal{M}$ = isometric embeddings), $T(V) = V^{**}$, $T$-dense (resp., $T$-closed) maps correspond to dense (resp., closed isometric) embeddings. The $T$-completion monad $\widetilde{T}$ is the Banach space (Cauchy) completion (Lucyshyn-Wright, 2014).
• Sheafification: For a topos $\mathcal{X}$ with Lawvere–Tierney topology $j$, $T$-density and $T$-closure correspond to $j$-dense and $j$-closed monomorphisms. The $T$-completion is the usual $j$-sheafification functor, and the completed objects are $j$-sheaves.

These examples demonstrate the unification of completion phenomena across analysis, topology, and algebraic geometry.

7. Outlook and Further Directions

Continuous relative completion is robust for transporting and controlling homological or cohomological properties across settings. It enables transfer of finiteness and cohomology results from discrete to non-discrete topological groups, equips period domains and Selmer stacks in arithmetic geometry with nontrivial towers, and yields mixed Hodge structures and period morphisms connecting topological and algebraic realizations. Open directions include the behavior of $L^2$-invariants under completion, interplay with profinite and étale completions, and further instances in motivic and quantum field theoretic contexts (Lucyshyn-Wright, 2014, Bonn et al., 2024, Tapušković, 2023, Luo, 2018, Kantor, 2020).