Monoidal monad

<p>In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a branch of mathematics, a monoidal monad 
  
    
      
        (
        T
        ,
        η
        ,
        μ
        ,
        
          T
          
            A
            ,
            B
          
        
        ,
        
          T
          
            0
          
        
        )
      
    
    {\displaystyle (T,\eta ,\mu ,T_{A,B},T_{0})}
  
 is a <a href="/facts/Monad_(category_theory)/6fX10Qrx">monad</a> 
  
    
      
        (
        T
        ,
        η
        ,
        μ
        )
      
    
    {\displaystyle (T,\eta ,\mu )}
  
 on a <a href="/facts/Monoidal_category/F2l5oKMD">monoidal category</a> 
  
    
      
        (
        C
        ,
        ⊗
        ,
        I
        )
      
    
    {\displaystyle (C,\otimes ,I)}
  
 such that the functor 
  
    
      
        T
        :
        (
        C
        ,
        ⊗
        ,
        I
        )
        →
        (
        C
        ,
        ⊗
        ,
        I
        )
      
    
    {\displaystyle T:(C,\otimes ,I)\to (C,\otimes ,I)}
  
 is a <a href="/facts/Lax_monoidal_functor/18z7djqP">lax monoidal functor</a> and the natural transformations 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
 and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 are <a href="/facts/Monoidal_natural_transformation/CDAeQNLG">monoidal natural transformations</a>. In other words, 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is equipped with coherence maps 
  
    
      
        
          T
          
            A
            ,
            B
          
        
        :
        T
        A
        ⊗
        T
        B
        →
        T
        (
        A
        ⊗
        B
        )
      
    
    {\displaystyle T_{A,B}:TA\otimes TB\to T(A\otimes B)}
  
 and 
  
    
      
        
          T
          
            0
          
        
        :
        I
        →
        T
        I
      
    
    {\displaystyle T_{0}:I\to TI}
  
 satisfying <a href="/facts/Lax_monoidal_functor/18z7djqP">certain properties</a> (again: they are lax monoidal), and the unit 
  
    
      
        η
        :
        i
        d
        ⇒
        T
      
    
    {\displaystyle \eta :id\Rightarrow T}
  
 and multiplication 
  
    
      
        μ
        :
        
          T
          
            2
          
        
        ⇒
        T
      
    
    {\displaystyle \mu :T^{2}\Rightarrow T}
  
 are <a href="/facts/Monoidal_natural_transformation/CDAeQNLG">monoidal natural transformations</a>. By monoidality of 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
, the morphisms 
  
    
      
        
          T
          
            0
          
        
      
    
    {\displaystyle T_{0}}
  
 and 
  
    
      
        
          η
          
            I
          
        
      
    
    {\displaystyle \eta _{I}}
  
 are necessarily equal.
</p><p>All of the above can be compressed into the statement that a monoidal monad is a monad in the <a href="/facts/Strict_2-category/YeMVfoAc">2-category</a> 
  
    
      
        
          
            M
            o
            n
            C
            a
            t
          
        
      
    
    {\displaystyle {\mathsf {MonCat}}}
  
 of monoidal categories, lax monoidal functors, and monoidal natural transformations.
</p>

Monoidal monad open-in-new

Monoidal monad