Zoran Majkić - Category Theory: Invariances and Symmetries in Computer Science

Comma-propagation is a general transformation applied to n-dimensional levels, their functors and natural transformations. In what follows, the operation of the comma-propagation will be denoted by {_} , and hence applied to a category C, functor F and natural transformation will be denoted by {C} , {F} and {} relatively. In this section, we will introduce this operation for the n-dimensional levels by the following definition.

If we consider the universal properties of a base category as (co)universal arrows as a kind of Lagrangian in category theory, and derived from them adjunctions and (co)limits, then we will consider their invariance under these comma-propagation transformations: a global symmetry in this case means that these categorial universal structures are preserved under these general transformations in all n-dimensional levels.

Let us show that each (n+1) -dimensional level Cn+1=CnCn=(CC)n , for n1 , with C1=C , of a given base category C, can be equivalently represented by the category of functors CnJ where Cn is n-dimensional level and the small index category J is equal to the preorder category with two objects a1=0 and a2=1 and the unique nonidentity arrow l12:a1a2 (representing partial order 01).

In fact, for an arrow (natural transformation) :FF in Cn2 between two objects F and F , which are functors F,F:2Cn , we have the corresponding arrow ((a1);(a2)):J(F(l12))J(F(l12)) in Cn+1=(CC)n , corresponding to the following commutative diagram in Cn : Vice versa, for each arrow (h;k):J(f)J(g) in Cn+1 , we have the arrow (natural transformation) :FF in Cn2 with components (a1)=h and (a2)=k , between functor F defined by F(l12)f and functor F defined by F(l12)g . Thus, Cn+1 is equivalent to Cn2 .

Because of that, the comma induction is discussed in the previous chapter as a particular case of functors between the category of functors CnJ and certain n-dimensional levels.

Consequently, the comma-propagated functors, natural transformation and (more generally) the concept of the comma-propagation, which we introduce in this section, generalize the comma-induced functors and natural transformations, by allowing that J be any small index category. It is well known that such categories J with a category of functors CJ are used to define the limits in the base category C, and this is the main reason for the introduction of comma-propagation, to see how the limits in this base category C1=C are inductively propagated in all higher n-dimensional levels Cn .

In fact, we can generalize the specific (for arrow categories) arrow diagonal functor :DDD and the standard diagonal functor :DDD and properties () in the Appendix, into a more general diagonal functor (ascending case) :DCJ , where C=DmDDm is a n-ary product of categories for finite m1 , as follows.

The following scheme with the commutative diagram demonstrates the relationship between the arrow-diagonal functor :D(DD) and this more general diagonal functor :DCJ , for m=1 , C=D1=D ,

Note that in the case when the small index category J=2 is composed by only two objects 0 and 1 and the nonidentity arrow e:01 , in the commutative diagram above (for j=1 and k=2 ) we have that aj=a1=0 , ak=a2=1 and arrow ljk=l12=e , so we have the exact correspondence between the objects 0(c) and J(idc) and arrows 1(g) and (g;g) in the categories D2 and (DD) , respectively.

The following scheme demonstrates the relationship between the standard diagonal functor :DDD and this more general diagonal functor :DCJ , in the case when m=2 , C=D2=DD and J=1 is the category of only one object a1 and its identity arrow ida1 ,

With this generalization, in the place of the duality functor L in (), used for the comma-induction, here for the more general comma-propagation we will introduce two new specific functors, called modulators, L and K as follows.