Soft Topological Groups

• Soft topological groups are structures that integrate group operations with soft topologies, parameterized by sets, to extend classical group theory.
• Morphisms in STGrp consist of parameter-set functions paired with soft-continuous group homomorphisms, ensuring coherent algebraic and topological interactions.
• The categorical framework of STGrp, equipped with a symmetric monoidal soft-product, enables innovative approaches to soft-connectedness and parameterized topology.

A soft topological group generalizes both group and topological structures to the setting of soft set theory by equipping a group with a “soft topology” parameterized by a set, and requiring the group operations to be soft-continuous. The category STGrp comprises soft topological groups as objects and pairs of parameter-set functions with soft-continuous group homomorphisms as morphisms. The category admits a symmetric monoidal structure via a soft-product operation, with coherence inherited from Set and Group. This framework provides a categorical foundation for the interplay of group-theoretic and soft-topological constructions (Alemdar et al., 16 Nov 2025).

1. Foundations of Soft Topological Groups

A soft set over a universe $X$ with parameter set $\xi$ is a map $W:\xi\to\mathcal{P}(X)$. A soft topology $\zeta$ on $G$ is a family of soft sets over $G$ (with parameter set $\xi$) satisfying soft analogues of topology axioms. A soft topological group $(G, \zeta)_\xi$ consists of a group $(G,\ast)$, a soft topology $\zeta$ on $G$, and a parameter set $\xi$, such that:

• The group multiplication $\ast\colon G\times G \to G$ is soft-continuous for the soft-product topology $\zeta_{\times}$.
• The inversion $\iota\colon G\to G,\, a\mapsto a^{-1}$ is soft-continuous.

These conditions are formally equivalent to the requirement: for any $a, b\in G$ and any soft open $W\ni ab^{-1}$, there are soft opens $U\ni a$, $V\ni b$ with $U\ast V^{-1}\,\widetilde{\sqsubseteq}\,W$.

2. Morphisms in the Category STGrp

Morphisms in STGrp are soft-continuous group-homomorphism pairs. Specifically, for soft topological groups $(G,\zeta)_\xi$, $(G',\zeta')_{\xi'}$, a morphism

$$(\varphi, \varrho): (G, \zeta)_\xi \to (G', \zeta')_{\xi'}$$

consists of:

• A function $\varphi : \xi \to \xi'$ on parameter sets,
• A group homomorphism $\varrho: G \to G'$,

with the property of soft continuity: for every $x\in G$ and $W' \in \zeta'$ with $\varrho(x) \in W'$, there exists $W \in \zeta$ with $x \in W$ and $\varrho(W)\,\widetilde{\sqsubseteq}\, W'$.

Component	Notation	Description
Parameter map	$\varphi: \xi \to \xi'$	Function on parameter sets
Homomorphism	$\varrho: G \to G'$	Group homomorphism
Soft continuity	$\varrho(W)\,\widetilde{\sqsubseteq}\, W'$	Soft-continuity condition

3. Category Structure and Laws

STGrp forms a category with:

• Objects: Soft topological groups $(G, \zeta)_\xi$.
• Morphisms: Soft-continuous pairs $(\varphi, \varrho)$.
• Composition: $$(\varphi_2, \varrho_2)\circ (\varphi_1, \varrho_1) = (\varphi_2\circ \varphi_1,\, \varrho_2\circ \varrho_1)$$.
• Identity morphisms: The pair $(\mathrm{id}_\xi, \mathrm{id}_G)$ for every object.
• Associativity and unity: Inherited directly from function and group-homomorphism composition.

These structures guarantee that diagrams involving morphisms and compositions in STGrp commute according to the axioms of category theory (Alemdar et al., 16 Nov 2025).

4. Monoidal Structure and Coherence

The categorical product in STGrp is given by the soft-product of soft topological groups. For $(G_1, \zeta_1)_{\xi_1}$, $(G_2, \zeta_2)_{\xi_2}$: $$(G_1, \zeta_1)_{\xi_1} \otimes (G_2, \zeta_2)_{\xi_2} = (G_1\times G_2,\, \zeta_{\times})_{\xi_1\times \xi_2}$$ where $\zeta_{\times}$ is generated by sets $U_1\widetilde{\times} U_2$ with $$(U_1\widetilde{\times} U_2)(e_1,e_2) = U_1(e_1)\times U_2(e_2)$$.

• Unit: The terminal object is $I = (\{1\},\{\emptyset, \{1\}\})_{\{*\}}$, the trivial one-point group with the indiscrete soft topology.
• Tensor on morphisms: Given morphisms $(\varphi, \varrho)$, $(\psi, \sigma)$, their tensor is $(\varphi\times \psi,\, \varrho\times \sigma)$.
• Coherence isomorphisms:
  • Associator: $\alpha_{1,2,3}$ rebrackets products and parameter sets.
  • Left/right unitors: $\lambda_G, \rho_G$ are projections.
  • Braiding: $\beta_{G,H}$ swaps both group and parameter components.
• Symmetry and coherence: The associator, unitors, and braiding satisfy pentagon, triangle, hexagon, and symmetry-involution diagrams as per Mac Lane’s coherence theorems.

This structure renders STGrp a symmetric monoidal category, with all coherence inherited from Set$^2$ via the Cartesian product (Alemdar et al., 16 Nov 2025).

5. Concrete Example: Soft Usual Topological Group on $\mathbb{R}$

An explicit example is constructed as follows:

• Underlying group: $\mathbb{R}$ under addition.
• Parameter set: Any nonempty set $\xi$.
• Soft topology: $\mathscr{U}_\xi$ with subbase $$\{ (e, (a,\infty)), (e, (-\infty, b)) \mid e\in\xi,\, a,b\in\mathbb{R} \}$$.
• Group operations: The addition map $\mathbb{R}^2\to\mathbb{R}$, $(x,y)\mapsto x+y$, and inversion $(x,y)\mapsto(-x,-y)$ are soft-continuous.

The soft tensor product $$(\mathbb{R}, \mathscr{U}_\xi)_\xi \otimes (\mathbb{R}, \mathscr{U}_\xi)_\xi$$ yields $$(\mathbb{R}^2, \mathscr{U}_\xi\times \mathscr{U}_\xi)_{\xi\times\xi}$$. Basic soft opens have the explicit form $$H^{U\times V}_{\xi\times\xi} = \{ ((e_1, e_2), U(e_1)\times V(e_2)) \mid (e_1, e_2)\in \xi\times\xi\}$$ for usual opens $U, V$.

This construction confirms that all category and monoidal operations restrict in natural ways, with associator and braiding given by the usual set and group product maps (Alemdar et al., 16 Nov 2025).

6. Structural and Categorical Properties

The category STGrp possesses robust categorical features:

• Admits finite products and a terminal object.
• Monoidal product is given by the soft-product of groups and soft topologies.
• Symmetry and all coherence properties are inherited strictly from the underlying categories of sets and groups, due to strict functoriality in each variable.
• The category structure and monoidal product are constructed so that all required categorical diagrams commute automatically.

A plausible implication is that the categorical framework established by STGrp may facilitate development of further algebraic and homotopical structures within soft set theory by analogy with classical topological group theory.

7. Significance and Further Developments

The categorical foundation of soft topological groups enables the systematic study of soft analogues of concepts such as connectedness, path connectedness, and continuous group actions. The preservation of soft connectedness properties under soft-continuous morphisms, as shown in this context, demonstrates the naturality of the STGrp formalism for both algebraic and topological soft structures. This suggests that STGrp provides a foundation for generalizing methods from topological groups to parameterized and uncertain data settings, with the potential for future extensions to soft algebraic topology, homological constructions, and applications where classical set-theoretic structure is insufficient (Alemdar et al., 16 Nov 2025).