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 [to do] fully iterative definition of goodstein HOS MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 07/17/2022, 11:51 PM Before I forget it, let me post here a note for the future. As defined formally by me elsewhere (document in preparation), formal in the sense of deprived of interpretation/representation as endofunction, a formal (pre-)Goodstein sequence inside a pointed non-commutative monoid $$(M,s)$$ with ranks belonging to an $$\mathbb N$$-iteration $$(J,{(-)}^+)$$ is a function $${\bf h}:J\to M$$ satisfying the system of $$\mathbb N$$-equivariance condition (aka superfucntion equations) $${\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}$$ Since after the latest discussions on the forum I have started to internalize and understand fully the relationship between being a superfucntion, being a family of superfunctions and being an iteration I believe that the previous version of the Goodstein f.equation shows itself as just the $$\mathbb N$$-equivariant version of a more general $$A$$-equivariant Goodstein f.equation. Definition ($$A$$-equivariant (pre-)Goodstein equation): Fix the monoid of time $$A$$ and an unit of time $$u\in A$$. Take an $$A$$-pointed non-comm. monoid $$(M,s)$$, it will be our support and monoid morphism $$s:A\to M$$ will be called the seed. Let $$(J,{(-)}^+)$$ be a an $$\mathbb N$$-action, called the space of ranks. An $$A$$-equivariant (pre-)Goodstein map is a map $${\bf h}:J\to {\rm Hom}_{\rm Mon}(A,M)$$, i.e. a sequence of $$A$$-iterations/monoid homomorphisms $${\bf h}_j:A\to M$$, over $$M$$ indexed by $$J$$ that satisfies the $$A$$-equivariant Goodstein f.equation over the seed $$s$$ wrt the unit of time $$u$$: $$\forall a\in A.\,{\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)$$ Example: look for the special case $$A=\mathbb R$$ adn $$u=1$$, then this means the $$\mathbb R$$-equivariant goodstein functional equation doesn't ask the next hyperoperation to be just a superfunction of the previous but also to respect $$\mathbb R$$-iterations of the previous. This means the new definition is more close to the naive expectation of what we would like Goodstein hyperoperations to be. This means we have a sequence of $$\mathbb R$$-iterations $$f_j^t$$ and that $$f^{\circ 1}_{j^+}\circ s^{\circ a}=f^{\circ a}_{j}\circ f^{\circ 1}_{j^+}$$ Open problem. some can clearly see that this is not perfection. The rank variable still belongs to the world of $$\mathbb N$$-iterations/actions. The ultimate Goodstein functional equation should be $$B$$-equivariant also i the rank variable... but how? The only way I can think of is by iterating group conjugation. We need to ask $$M$$ to be a group. In this way, maybe we can find to make $$A$$-equivariant (pre-)Goodstein map $${\bf h}:J\to {\rm Hom}_{\rm Mon}(A,M)$$ into a $$B$$-equivariant map, for some monoid $$B$$ acting on the space of ranks.... but how? MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 07/18/2022, 05:40 PM (This post was last modified: 07/22/2022, 01:11 AM by MphLee.) Additional note. the problem of extending $$B$$-equivariance to the space of ranks need to be analyzed as follows. We need to keep track of where our maps lives. There are six cases of increasing complexity $$\begin{array}[|ccc|cc|] &&&&{\rm ranks\, equiv.}&\\ \hline {\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying\, goodstein} &B=1& A=\mathbb N\\ {\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=1& A\\ \hline \mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B=\mathbb N&A=\mathbb N\\ \mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=\mathbb N&A\\ \hline B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B&A=\mathbb N\\ B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B&A\\ \hline \end{array}$$ Classical case over monoids. In the case of $$M$$ a monoid $$\mathbb N$$-equivariant goodstein map is a map that lives in the category $$1{\rm -Act}={\rm Set}$$ $${\bf h}\in {\rm Hom}_{\rm Set}(J,{\rm Hom}_{\rm Mon}(\mathbb N,M))$$ that satisfies additional condition $${\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}$$, i.e. each $${\bf h}_j$$ is a map in $$\mathbb N{\rm -Act}={\rm Set}^{B\mathbb N}$$. Equivariant case over monoids. In the case the case of $$A$$-equivariant goodstein maps is a map that still lives in $$1{\rm -Act}={\rm Set}$$ $${\bf h}\in {\rm Hom}_{\rm Set}(J,{\rm Hom}_{\rm Mon}(A,M))$$ but this time they satisfies $$\forall a\in A.\, {\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)$$, i.e. each $${\bf h}_j$$ is a map in $$A{\rm -Act}={\rm Set}^{BA}$$. In the case of $$M=G$$ a group an incredible simplification becomes available for $$\mathbb N$$-equivariant goodstein maps: since $${\rm Hom}_{\rm Mon}(\mathbb N,G)\simeq G$$ and on this set we can define the subfunction map $$\Sigma_s:G\to G$$, sending $$g\in G\mapsto{ }gsg^{-1}$$ we can upgrade the object $${\rm Hom}_{\rm Mon}(\mathbb N,G)$$ from the category $$1{\rm -Act}={\rm Set}$$ making it into an object $$\Sigma^G_S:=(G,\Sigma_s)$$ of  $$\mathbb N{\rm -Act}$$, the same place where the space of ranks $$J=(J,(-)^-)$$ lives. It turns out that the goodstein can be sinthetized by asking for maps that live in the category $$\mathbb N{\rm -Act}$$. Classical case over groups. In the case of $$M=G$$ a group, a $$\mathbb N$$-equivariant goodstein map is just map that lives in the category $$\mathbb N{\rm -Act}$$ of discrete dynamical systems $${\bf h}\in {\rm Hom}_{\mathbb N{\rm -Act}}(J,\Sigma_s^G)$$ Nothing more should be asked! This already implies it satisfies $${\bf h}_{j}s={\bf h}_{j^-}{\bf h}_{j}$$. Equivariant case over groups. Here we have a big obstacle since $${\rm Hom}_{\rm Mon}(A,G)$$ is not equivalent to $$G$$ in general and on this space of group homorphisms is not closed under the map $$\Sigma_s$$ unless $$G$$ is abelian... but in this case everything becomes trivial: even if it was it is not clear this is what we need to enforce the equivariant goodstein condition. The question is: what is the endofunction over $${\rm Hom}_{\rm Mon}(A,G)$$ that we should consider and why? This is a big problem. Maybe we can face this from a synthetic point of view. We need an object $$\mathfrak S(A,G)\in \mathbb N{\rm -Act}$$ of the category  $$\mathbb N{\rm -Act}$$ associated with the group $$G$$ that behaves as if it were the space of $$A$$-iterations over $$G$$ and closed under taking subfunctions... something that is not possible as stated. We then look for maps $${\bf h}\in {\rm Hom}_{\mathbb N{\rm -Act}}(J,\mathfrak S(A,G) )$$ And such that, given enough extra structure, we can somehow reconstruct the condition $$\forall a\in A. \Sigma_{s(a)}({\bf h}_{j}(u))={\bf h}_{j^-}(a)$$. Two more cases are to be studied and that completes the study of goodstein maps: the case with equivariant ranks but classical maps, and the case with everything equivariant. $$B$$-Equivariant ranks and classical Goodstein over group. The first is again pretty straightforward. Just take any $$B$$-iteration of the map $$\Sigma_s:G\to G$$, i.e. a map $${\boldsymbol \Sigma}:B\to G^G$$ s.t. $$\forall \alpha,\beta\in B.\, \, {\boldsymbol \Sigma}(\alpha+\beta,g)={\boldsymbol \Sigma}(\alpha,{\boldsymbol \Sigma}(\beta,g))$$ s.t. for some $$\upsilon \in B$$ we have $${\boldsymbol \Sigma}(\upsilon,g)=gsg^{-1}$$. And as space of ranks take an object $$J\in B{\rm -Act}$$ We look for maps $${\bf h}\in {\rm Hom}_{B{\rm -Act}}(J,{\boldsymbol \Sigma})$$ This is actually iterating conjugation... it is really the way to access authentic non-integer ranks as started by Trappmann in his 2007 2008 thread: non-natural operation ranks. But is hard... and we really should be interested when $$M$$ is not a group... thus making everything harder. $$B$$-Equivariant ranks and $$A$$-equivariant Goodstein over group. Here the model becomes highly non-trivial... I still don't know how to treat this... Probably the way to go is to upgrade this discussion, that is really at set theoretic level to categories. If we turn every object into a category... and every map into a functor... maybe all the problem presented here will dissolve. MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 07/19/2022, 02:11 PM Let's switch to the symbolic dynamic version, aka the shift goodstein condition $$\sigma({\bf h})\cdot s={\bf h}\cdot \sigma({\bf h})$$. Maybe we the right way to go is by obtaining the $$A$$-equivariant shift goodstein as a normal equation inside the structure $${\rm Hom}_{\rm Mon}(A,M)^J$$: $${\rm ev}_u(\sigma({\bf h}))\cdot s={\bf h}\cdot {\rm ev}_u(\sigma({\bf h}))$$. A quick check $$({\rm ev}_u(\sigma({\bf h}))\cdot s)_j=({\rm ev}_u(\sigma({\bf h})))_j\cdot s_j$$ the problem is that $$({\rm ev}_u(\sigma({\bf h})))_j$$ is not defined in any meaningful way unless we move to another codomain by transporting the equation along the function $$u^*: {\rm Hom}_{\rm Mon}(A,M)^J\to{\rm Hom}_{\rm Mon}(\mathbb N,M)^J$$ but then it just is a rephrasing of the previous definition that doesn't add anything of value... it seems $$\forall a\in A.\, u^*(\sigma({\bf h}))\cdot a^*(s)=a^*({\bf h})\cdot u^*(\sigma({\bf h}))$$ or maybe isn't... since there is a canonical map $$EV({\bf x},a)=a^*({\bf x})$$ $$EV: {\rm Hom}_{\rm Mon}(A,M)^J\times A\to{\rm Hom}_{\rm Mon}(\mathbb N,M)^J$$ define then for $$M=G$$ the map $$\mathcal F ({\bf h}, a)=EV(\sigma{\bf h},u)\cdot EV(s,a)\cdot EV(\sigma{\bf h},u)^{-1}$$ and ask for $$\bf h$$ solutions of $$\forall a.\,\mathcal F({\bf h},a)=EV({\bf h},a)$$ Apply curry to the second variable, obtain that $$A$$-equivariant goodstein maps are solutions of $${\bar {\mathcal F}}({\bf h})=\bar{EV}({\bf h})$$ MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 07/21/2022, 07:10 PM The last attempt seems too complicated to give low hanging fruits I think it is better to start from this table $$\begin{array}[|ccc|cc|] &&&&{\rm ranks\, equiv.}&\\ \hline {\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying\, goodstein} &B=1& A=\mathbb N\\ {\rm set\, theoretic\,}{\bf h}&over\, monoid\, M&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=1& A\\ \hline \mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B=\mathbb N&A=\mathbb N\\ \mathbb N{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B=\mathbb N&A\\ \hline B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying\, goodstein} &B&A=\mathbb N\\ B{\rm -Act}\, {\rm theoretic\,}{\bf h}&over\, group\, G&{\rm satisfying}\, A{\rm -equivariant\, goodstein} &B&A\\ \hline \end{array}$$ and refine it to a big picture. Here the general scheme is: we take a map inside a certain category, from an object of ranks to an object of the support, and ask it to satisfy a condition, be it goodstein condition or some equivariant goodstein, or variant of it. All of this must be fixed by appropriate forgetfull functors $$U$$ sending objects to the appropriate categories. From now on I'll identify the categories $$A{\rm - Act}$$ and the functor (co-presheaves) category $$[A,{\rm Set}]$$ whenever $$A$$ is a monoid and I'll omit the $$B$$-functor in order to keep notation clear. I'll use the common practice of denoting the hom-sets of a category $$\mathcal C$$ by $$\mathcal C(X,Y)$$ instead of $${\rm Hom}_{\mathcal C}(X,Y)$$ except in the case of sets. Set theoretic goodstein maps This time I'll break the scheme differently, leaving aside everything that is $$A$$-equivariant since it need a qualitative upgrade that is intermediate to going categorical. $$\begin{array}[|cccc|] f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\ \hline {\bf h}\in{\rm Hom}(UJ,UM)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{+}{\to}J)\in \mathbb N{\rm - Act}& M\in{\rm Mon}&&\forall j\in UJ.\, {\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}&{\bf goodstein\, equation\,\, (GE)}\\ {\bf h}\in{\rm Hom}(UJ,UM)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{-}{\to}J)\in \mathbb N{\rm - Act}& M\in{\rm Mon}&&\forall j\in UJ.\, {\bf h}_{j}s={\bf h}_{j^-}{\bf h}_{j}&{\bf anti}\!-\!{\rm GE\,\, (GE^-)}\\ {\bf h}\in{\rm Hom}(UJ,UG)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{+}{\to}J)\in \mathbb N{\rm - Act}& G\in{\rm Grp}&&{\bf h }\in {\rm fix}((\Sigma^G_S)_* \circ (+)^* )&{\bf GE\,\, over\, grps}\\ {\bf h}\in{\rm Hom}(UJ,UG)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{-}{\to}J)\in \mathbb N{\rm - Act}& G\in{\rm Grp}&&\forall j\in UJ.\, \Sigma^G_s({\bf h}_{j}^{-1})={\bf h}_{j^-}&{\bf GE^-\,\, over\, grps}\\ \hline \end{array}$$ Special cases. since we just need monoid and groups that are non-abelian we can study set theoretic goodstein map on nice groups/monoids to obtain particular subtheories where is easier to obtain explicit results/computations. I believe I have proven that the theory over finite groups is trivial, but the problem is open over finite monoids. The theory over the  monoid $$\mathbb N^\mathbb N$$ is basically extends recursion theory and gives hyperoperations. Also we can investigate the theory over multiplicative groups or rings or $$R$$ algebras, i.e. $$R$$-modules $$X$$ equipped with bilinear unital/associative maps $$(-,-)_X:X\times X\to X$$. Or, the most interesting case, by multiplication arising as multiplication in general linear groups of $$n$$-order matrices over a field (eg. finite fields). Here four interesting special cases that seems promising. $$\begin{array}[|cccc|] f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\ \hline A\in{\rm Hom}(U\mathbb N,{\rm End}(\mathbb N))&{in\,\,}1{\rm -Act} \,(of\,sets) & (\mathbb N\overset{S}{\to}\mathbb N)\in \mathbb N{\rm - Act}& \mathbb N\in{\rm Set}&&A(n+1,x+1)=A(n,A(n+1,x))&{\bf Ackermann\, equation\,\,}\\ {\bf f}\in{\rm Hom}(UJ,UX)&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{+}{\to}J)\in \mathbb N{\rm - Act}& X\in R{\rm -Alg}&& ({\bf f}_{j},s)_X=({\bf f}_{j},{\bf h}_{j^+})_X&\\ f\in{\rm Hom}(UJ,R^{\times})&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{+}{\to}J)\in \mathbb N{\rm -Act}& R\in{\rm Ring}&&f(j^+)\cdot s=f(j)\cdot f(j^+)&{\bf GE\,\, over\, non-comm.\,rings}\\ M\in{\rm Hom}(UJ,U{\sf GL}_n(k))&{in\,\,}1{\rm -Act} \,(of\,sets) & (J\overset{+}{\to}J)\in \mathbb N{\rm -Act}& k\in{\rm Field}&& M_{j^+}SM_{j^+}^{-1}=M_j&k{\bf -linear\, GE\,\,}\\ \hline \end{array}$$ The set theoretic approach seems very rich yet limited. It boils down to these 4 cases, they can be reduced by two if the ranks dynamics is invertible because at that point goodstein and anti-goodstein equations give same solutions. Dynamical goodstein maps The first generalization appears spontaneously if we restrict the previous theory to groups. Before generalizing further. Notice that the support object needs to induce a monoid operation over the Hom-set so we are pretty limited in extending the category from where we pick the support object. We are also limited in the choice of the ranks object. It can't just be a set because the goodstein equation ask us for a procedure that gives "the next rank", so it has to be at least a discrete dynamical system, or something equipped with an action of something that can be restricted to a discrete action: an object of a category equipped with an appropriate forgetfull functor $$F:\mathcal C\to\mathbb N{\rm -Act}$$. Generalizing. as I've noticed before, groups makes a cool phenomenon to appers. The anti-goodstein one does reduce in such a way that we can make it hold by structural means, without asking for it. We then use instead of the forgetfull functor, the functor $$(G,s\in G) \mapsto \Sigma^G_s$$ from pointed groups to dynamical systems sending a group and an element to the sub-function by s (it is functorial in $$(G,s)$$). This way we can naturally extend the scheme to $${\mathbb N}$$-equivariant goodstein maps with $$B$$-equivariant ranks. We take the ranks to be equipped with a $$B$$-action $$J$$, for $$B$$ a monoid equipped with an unit of time $$\upsilon\in B$$, and as support object we take a $$B$$-action over the group $$G$$ that when restricted to the natural numbers gives back $$\Sigma^G_s$$: it is an element of the preimage category $${\boldsymbol \Sigma}\in (\upsilon^*)^{-1}(\Sigma^G_s)\subseteq B{\rm -Act}$$ also expressible as the pullback, i.e. the fiber of the restriction functor bundle $$\upsilon^*$$ that we can denote more comfortably as $$B{\rm -Act}_{\Sigma^G_s}$$. $$\begin{array}[|cccc|] f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\ \hline {\bf h}\in\mathbb N{\rm -Act}(J,\Sigma^G_s)&{in\,\,}\mathbb N{\rm -Act} & J\in B{\rm -Act}& G\in{\rm Grp}&¬hing&{\bf GE^-\,\,}by\, default\\ {\bf h}\in B{\rm -Act}(J,{\boldsymbol \Sigma})&{in\,\,}\mathbb N{\rm -Act} & J\in B{\rm -Act}& {\boldsymbol \Sigma}\in B{\rm -Act}_{\Sigma^G_s}&&i.e. \, \forall \beta \in B,\,j\in J.\, {\bf h}_{\beta j}={\boldsymbol \Sigma}^\beta ({\bf h}_j)&B{\bf -equiv.\, ranks\, GE^-\,\,}by\, default\\ \hline \end{array}$$ This route seems even more promising but is much more limited than the previous one... I believe it is a dead end or maybe something bringing us to a totally different theory: the theory of group conjugation and its meaning. I believe we should go another way and turning everything categorical. Set-theoretic $$A$$-equivariant-goodstein maps Here is the point where things gets harder at first but I've seen an opening for going functorial. Let $$A$$ be a monoid and $$u\in A$$ the unit of time. The concept is simple: goodstein equation impose pointwise the condition of being $$\mathbb N$$-equivariant. We use the abstract identification of points $$x\in M$$ of the support monoid to $$\mathbb N$$-iterations over $$M$$, i.e. $$M\simeq {\rm Mon}(\mathbb N,M)$$ and we use the same philosophy for seeing $$A$$-equivariant goodstein maps as selecting many $$A$$-iterations over $$M$$ that by the $$A$$-equivariant goodstein equation ($$A$$-EGE) have their $$A$$ equivariance imposed pointwise. $$\begin{array}[|cccc|] f{\rm maps\, in}&{\bf category}&{from\,the\,\,{\bf ranks}\,\, object}&{to\,the\,\,{\bf support}\,\, object}&&{\rm satisfying}&\\ \hline {\bf h}\in {\rm Hom}(UJ,{\rm Mon}(\mathbb N,M))&{in\,\,}1{\rm -Act} & J\in \mathbb N{\rm -Act}& M\in{\rm Mon}&&\forall j\in UJ.\, {\bf h}_{j^+}s={\bf h}_{j}{\bf h}_{j^+}&{\bf GE\,\,}\\ {\bf h}\in {\rm Hom}(UJ,{\rm Mon}(A,M))&{in\,\,}1{\rm -Act} & J\in \mathbb N{\rm -Act}& A,M\in{\rm Mon}&&i.e. \, \forall j\in UJ,\,a\in A.\, {\bf h}_{j^+}(u)s(a)={\bf h}_{j}(a){\bf h}_{j^+}(u)&A{\bf -EquivGE\,\,}\\ \hline \end{array}$$ Road to category theoretic goodstein maps. Here it is where we can spot the opening. First, note that until now the most lazy of our parameters was the object of ranks but since we can send dynamical system to sets and sets to discrete categories, embedding the category of sets in the category of categories is all we need. Secondly, the category of monoids can be enriched in cat: this means that the set of monoid homorphims is itself a category, a functor category $$[A,M]$$: namely  $${\rm Ob}([A,M])={\rm Mon}(A,M)$$ The reason I think this happens is because monoids themselves can be seen as one-object categories, monoid morphims as functors and natural transformations between two functors $$f,g:A\to M$$ are exactly $$A$$-equivariances. I find this spectacular. In symbols: let $$M(x,y)=\{\phi\in M:\, \phi x=y\phi\}$$ then we have a bijection $$M(x,y)\simeq {\rm Nat}_{[\mathbb N,M]}(x^\bullet,y^\bullet)$$ where $$x^\bullet:\mathbb N\to A$$ is the monoid morphism defined as $$x^0=1_M$$ and $$x^{n+1}:=xx^n$$. In the same way, let $$f,g:A\to B$$ be functor between monoid seen as categories. natural transformations $$\phi:f\implies g$$ are elements $$\phi\in M$$ that satisfy $$\forall a.\, \phi f(a)=g(a)\phi$$... so if $$M^A(f,g)$$ is the set of such $$\phi$$s then we have a bijection $$M^A(f,g)\simeq {\rm Nat}_{[A,M]}(f,g)$$ The question remains... if we use functor categories as support objects... how we define on them the monoid operation on objects? t seems to be possible only when $$M$$ is abelian or for some special $$A$$ like the integers or the naturals. TO BE CONTINUED MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ « Next Oldest | Next Newest »

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