# A Note on the Soft Group Category
## 3 Soft Group Category  
The objects of the soft group category are soft groups, and the morphisms between these objects are soft group homomorphisms. The composition in this category is defined as the composition of soft group homomorphisms. **Proposition 1**. : _Soft groups and soft group homomorphisms between them form a category._  
Proof. : The proof is clear from the Equation (1) and Theorem 1. Note here that in this category for each object \((F,A)_{G}\), the unit morphism is defined as the soft group homomorphism \((1_{G},1_{A})\), where \(1_{G}:G\to G\) is the identity group homomorphism and \(1_{A}:A\to A\) is an identity map. Throughout this paper, this category is denoted by \(SGp\). We will give the definition of a soft kernel, which is not included in the literature and will be needed to prove that a monic morphism is one-to-one in the category of soft groups inspired by group theory. **Definition 10**. : _Let \((F,A)_{G}\) and \((H,B)_{K}\) be any two soft groups. Let \((f,p):(F,A)_{G}\to(H,B)_{K}\) be a soft group homomorphism. Let \(A^{\prime}\) be the set consisting of \(x\in A\) such that \(F(x)=Kerf\). Having fixed_  
\[A^{\prime}=\{x\in A:F(x)=Kerf\}, \tag{2}\]  
_we define \(F^{\prime}\) as the restriction function of \(F\) to \(A^{\prime}\). Then, the soft group \((F^{\prime},A^{\prime})_{G}\) is called the soft kernel of \((f,p)\)._  
**Remark 1**. : _The soft kernel of a soft group homomorphism \((f,p):(F,A)_{G}\to(H,B)_{K}\) may not always be defined. One can construct a soft group homomorphism \((f,p):(F,A)_{G}\to(H,B)_{K}\) with the soft kernel by defining \(F(x)=Kerf\) for some \(x\in A\)._  
**Theorem 2**. : _Let \((F,A)_{G}\) and \((H,B)_{K}\) be any two soft groups. Assume that \((f,p):(F,A)_{G}\to(H,B)_{K}\) is a soft group homomorphism such that \(F(x)=Kerf\) for some \(x\in A\). Then, \(f\) is injective if and only if \((F^{\prime},A^{\prime})_{G}\), which is the soft kernel of \((f,p)\), is the trivial soft subgroup._  
Proof. : Let \(f:G\to H\) be a monomorphism, and let \(F^{\prime}\) be the restriction of the function \(F\) to \(A^{\prime}\), where \(A^{\prime}\) is defined as in the Equation (2). Thus, the pair \((F^{\prime},A^{\prime})_{G}\) is the trivial soft subgroup of \((F,A)_{G}\). Conversely, let \((F^{\prime},A^{\prime})_{G}\) be the trivial soft subgroup, where \((F^{\prime},A^{\prime})_{G}\) is the soft kernel of \((f,p)\). We can write \(F^{\prime}(x)=Kerf=\{e\}\) for all \(x\in A^{\prime}\). Therefore, \(f\) is an injective group homomorphism. **Definition 11**. : _Let \((f,p)\) and \((g,q)\) be two soft group homomorphisms. Then, \((f,p)\) is equal to \((g,q)\) if and only if \(f=g\) and \(p=q\)._  
**Theorem 3**. : _Let \((f,p):(F,A)_{G}\to(H,B)_{K}\) be a soft group homomorphism such that \(F(x)=Kerf\) for some \(x\in A\). If \((f,p)\) is monic, then \(f\) and \(p\) are injective._  
**Proof. **  
Suppose that \(f\) is not injective. Thus, we have \(Kerf\
eq[e]\). When \((f,p)\circ(i_{Kerf},i_{A^{\prime}})=(f,p)\circ(f^{\prime},i_{A^{\prime}})\), we obtain \((i_{Kerf},i_{A^{\prime}})\
eq(f^{\prime},i_{A^{\prime}})\), where \(f^{\prime}:Kerf\to G,f^{\prime}(x)=e\). Hence the map \((f,p)\) is not monic. Now \(p\) is not injective. Then, we have two different \(x,y\in A\) such that \(p(x)=p(y)\). If we define the maps \(p_{1},p_{2}:A\to A\) as  
\[p_{1}(a)= \begin{cases}a,&a\in A-\{x,y\}\\ x,&a\
ot\in A-\{x,y\}\end{cases} \]  
and  
\[p_{2}(a)= \begin{cases}a,&a\in A-\{x,y\}\\ y,&a\
ot\in A-\{x,y\}\end{cases} \]  
then we have \(p\circ p_{1}=p\circ p_{2}\). Now, we want to show that the map \((1_{G},p_{1}):(F,A)\to(F,A)\) is a soft group homomorphism. In what follows, we need to prove that the diagram  
is commutative. Suppose \(a\in A-\{x,y\}\). Then, we obtain  
\[(F\circ p_{1})(a)=F(a)=\hat{1_{G}}(F(a))=(\hat{1_{G}}\circ F)(a).\]  
Due the fact that \(\hat{f}\circ F=H\circ p\) and \(f\) is injective, we conclude that \(\hat{f}(F(x))=H(p(x))=H(p(y))=\hat{f}(F(y))\) and so  
\[F(x)=F(y). \tag{3}\]  
Now, we first take \(a=x\).