Galois Semi-Covering Functor

• Galois Semi-Covering Functors are categorical mechanisms that generalize classical Galois theory by embracing nonfree group actions and relaxed covering conditions.
• They decompose Hom-sets in module and representation categories, preserving key invariants and structural properties across different mathematical settings.
• Their applications span topology, algebra, and higher category theory, enabling generalized correspondences and fruitful insights into monodromy and symmetry.

A Galois semi-covering functor arises as a categorical mechanism that generalizes the classical Galois correspondence to settings—topological, algebraic, and categorical—where the standard requirements for classical Galois theory (such as freeness of group actions or strict covering properties) are relaxed, yet enough structure remains to encode symmetries, monodromy, and invariants in functorial language. Such functors interpolate between Galois coverings and their analogues in settings including non-free group actions, ramified covers, non-locally connected spaces, skew group algebras, and representation-theoretic contexts involving finite groups, monoidal categories, and higher categorical structures.

1. General Framework and Formal Definition

A Galois semi-covering functor $F : \mathcal{C} \to \mathcal{D}$ is, in broad terms, a functor between categories (often module or representation categories) equipped with a group action (by a finite or profinite group $G$) such that $F$ encodes the symmetries or monodromy in a manner analogous to the classical Galois situation, but allowing for nontrivial stabilizers or other local pathologies.

In the setting of module categories over a $K$-algebra $\Lambda$ with a $K$-linear action of a finite group $G$, and $\Lambda G$ the corresponding skew group algebra, a Galois semi-covering functor $F: \Lambda\text{-mod} \to \Lambda G\text{-mod}$ admits decompositions of Hom-sets as follows (Sardar et al., 27 Jul 2025):

$$\mathrm{Hom}_{\Lambda G}(F(M), F(N)) \cong \begin{cases} \displaystyle\bigoplus_{g\in G} \mathrm{Hom}_\Lambda({}^g M,N) & \text{if } G_M \ne G \ \displaystyle\bigoplus_{g\in G} \mathrm{Hom}_\Lambda(M,{}^g N) & \text{if } G_N \ne G \ \mathrm{Hom}_\Lambda(M,N)^{|G|} & \text{if } G_M = G, G_N = G. \end{cases}$$

where ${}^g M$ denotes the twist of $M$ by $g$ and $G_M$, $G_N$ are the stabilizers of $M$, $N$. When the group action is free (all stabilizers trivial), $F$ is called a Galois covering functor; otherwise, it is only a semi-covering.

In categorical Galois theory, a "semi-covering" functor is broadly one that, while not necessarily covering all objects or preserving all covering conditions, still respects enough of the local or fibered structure to allow a Galois-type correspondence or monodromy formalism (Brazas, 2011, Berger et al., 2017, Pastuszak, 5 Mar 2025).

2. Semi-Topological Galois Theory and Splitting Coverings

The paradigm of the Galois semi-covering functor first emerges in the context of semi-topological Galois theory (1006.1166). Here, one studies Weierstrass polynomials $f_x(z)$ over a topological space $X$ and seeks a minimal covering $p: E_f \to X$ over which $f$ splits into linear factors with continuous root functions $a_1, \dots, a_n : E_f \to \mathbb{C}$. The assignment $f \mapsto E_f$, and further $$f \mapsto G_f := \operatorname{Aut}_{q^* C(X)}(q^* C(X)[a_1,\dots,a_n])$$, forms a functor from polynomials (with prescribed root behavior) to covering spaces and their automorphism groups.

This functor encapsulates a Galois semi-covering structure: the lattice of intermediate coverings between $E_f$ and $X$ is in bijection with the lattice of subgroups of the group of covering transformations $A(E_f/X)$, generalizing the classical Galois correspondence. The entire construction is functorial in the sense that morphisms (i.e., inclusions of subgroups or intermediate coverings) correspond under the functor [(1006.1166), Section 6].

3. Enriched Coverings and Galois Semi-Covering Functors in Topology

In the generalization to spaces that are not locally simply connected or locally path connected, such as the Hawaiian earring, classical covering theory fails. In "Semicoverings: a generalization of covering space theory" (Brazas, 2011), the fundamental groupoid is topologized (as $\pi^{\tau}X$), and a (semi)covering is classified by a continuous monodromy functor $\mathscr{M}p: \pi^{\tau} X \to \mathbf{Set}$. The semi-covering functor, in this context, sends spaces to the category of continuous monodromy functors.

The Galois semi-covering functor thus manifests as a functor from spaces (or suitable generalized spaces) to categories of continuous group(oid) actions or fibers, preserving the partial Galois correspondence: under suitable conditions, connected semi-coverings correspond to open subgroups of $\pi_1^{\tau}(X, x_0)$.

4. Semi-Coverings in Representation Theory and Skew Group Algebras

Let $\Lambda$ be a finite-dimensional algebra over an algebraically closed field $K$, and $G$ a finite abelian group of $K$-linear automorphisms, possibly with nonfree action. The functor $$F_\lambda: \mathrm{mod}\text{-}\Lambda \to \mathrm{mod}\text{-}\Lambda G$$ defined by

$$F_\lambda(M)(\bar{e}) = \bigoplus_{F(x)=\bar{e}} M(x),$$

(where the sum runs over representatives of the $G$-orbit) is exact, but not necessarily dense. The decomposition of Hom-sets and preservation—or splitting—of irreducible morphisms and almost split sequences reflects a Galois semi-covering structure [(Sardar et al., 27 Jul 2025), Theorem 3.5; Corollary 4.1].

When $G$ acts freely, this functor is a classic Galois covering. When the action has fixed points, beyond monodromy, the functor still "covers" the equivariant structure semi-globally: many invariants (ranks, stable ranks) remain preserved. The functor is "semi-dense" rather than dense, meaning some objects or morphisms are not hit, but a full subcategory (the modules of the first kind) is precisely the image (Pastuszak, 5 Mar 2025).

5. Categorical Galois (Semi-)Coverings and Functorial Adjoint Structures

In the context of locally bounded $K$-categories, a Galois covering functor $F : \mathcal{R} \to \mathcal{A} = \mathcal{R}/G$ (with $G$ acting freely on objects) induces a hierarchy of adjoint functors between module and functor categories (Pastuszak, 24 Feb 2025, Pastuszak, 5 Mar 2025).

• Pull-up: $$\Psi = (F_\lambda)_\bullet : \mathrm{MOD}(\mathcal{A}) \to \mathrm{MOD}(\mathcal{R})$$, defined by precomposition.
• Push-down/adjoin: left and right adjoints, $$\Phi, \Theta : \mathrm{MOD}(\mathcal{R}) \to \mathrm{MOD}(\mathcal{A})$$, which restrict to $\mathcal{F}(R)\to\mathcal{F}(A)$ for finitely presented functors.

If $F_\lambda$ is not dense (for example, due to fixed points or incomplete orbit coverage), these adjoints define a Galois precovering or semi-covering at the level of functor categories, with objects of the image known as functors "of the first kind" (Pastuszak, 24 Feb 2025). The semi-covering property is reflected in the non-denseness: $\mathcal{F}(A)$ splits into functors of the first and second kind.

This structure is crucial in comparing homological invariants, such as the Krull–Gabriel dimension ($KG(-)$):

$KG(\mathcal{R}) \le KG(\mathcal{A}),$

with equality often holding in the dense case; in the semi-dense (semi-covering) case, subtle differences may arise, and analysis of functors of the second kind becomes central.

6. Higher Auslander–Reiten Theory, Cluster Tilting, and Semi-Coverings

In higher homological frameworks, as in higher Auslander–Reiten theory, push-down functors under a $G$-action preserve substantial "cluster-tilting" structures. If $\mathcal{C}$ is a locally bounded, support-finite category with a free group action $G$, the pushdown functor

$$P_*: \mathrm{mod}\text{-}\mathcal{C} \to \mathrm{mod}\text{-}(\mathcal{C}/G)$$

maps $G$-equivariant $n$-precluster-tilting subcategories to $n$-precluster-tilting subcategories in the quotient, preserving higher AR-translation and Ext vanishing conditions. Conversely, inverse images reconstruct $G$-equivariant subcategories from the quotient. When the pushdown is not dense, the semi-covering language applies, capturing exactly which higher homological structures persist under relaxation of the action (Asadollahi et al., 19 Jun 2025).

7. Categorical Galois Connections and Semi-Covering Functors via Monads

Beyond the abelian or topological settings, "Galois semi-covering functor" has a formal counterpart in categorical Galois theory as established via monads and their invariants/stabilizers. Given an augmented monad $T$ with submonads $S$ and a forgetful functor $U$ from the Eilenberg–Moore category, one obtains a Galois connection (Vargas, 2021):

$$\mathrm{Sub}\,\text{monads}(T) \rightleftarrows \mathrm{Sub}\text{--functors}(U)$$

with invariants computed via right adjoints and stabilizers via Tannakian ends. The functoriality inherent in these correspondences provides a categorical template for what, in the context of field extensions, would be the classical field-subgroup lattice anti-isomorphism. The semi-covering language is appropriate when one has only partial stabilization or non-faithful covering.

8. Broader Implications and Role in Modern Theory

The Galois semi-covering functor underpins:

• The geometric encoding of monodromy and deck group symmetries in both topological and algebraic contexts, via splitting coverings or ramified Galois covers.
• The passage from an object with symmetry (via automorphisms or group actions) to a quotient or covering category, retaining functorial control over invariants, irreducibles, moduli, and almost split sequences.
• The functorial realization of "generalized" Galois correspondences, including in situations where ramification, non-freeness, or categorical complications preclude classical covering theory.

Semi-covering functors provide optimal frameworks for understanding moduli problems, structure of categories and algebras under group actions, and invariants in higher categorical or representation-theoretic settings. They underpin current approaches to the inverse Galois problem (realizing finite groups as Galois/monodromy groups in geometric contexts), the paper of moduli stacks and their components, and the translation of deep algebraic or topological invariants via categorical lenses.

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Table: Comparison of Covering vs. Semi-Covering Functors in Key Settings

Aspect	Galois Covering Functor	Galois Semi-Covering Functor
Action	Free group action	Nonfree, possibly with stabilizers
Dense functor	Yes	Not necessarily (semi-dense)
Hom-set decomposition	$\bigoplus_{g} \mathrm{Hom}_A({}^g X, Y)$	As in covering, but with possible blocks
Preservation of irreducibles	Always	Preserved unless both fixed
Application	Classical covering theory, torsors	Skew group algebras, ramified covers
Galois correspondence	Full lattice isomorphism	Partial or lattice-inverting assignment

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In all, a Galois semi-covering functor encodes weakened but still structurally rich relations between symmetry, monodromy, and descent in many modern mathematical settings—integrating techniques from topology, algebra, and category theory to generalize and extend the scope of classical Galois theory.