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http -> https (the question was bumped anyway)

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edited Sep 3 at 16:04

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The following family of examples can be extracted from Lenart and Ray, Some applications of incidence Hopf algebras to formal group theory and algebraic topology Some applications of incidence Hopf algebras to formal group theory and algebraic topology (Postscript). Let $B_2$, $B_3$, $B_4$, … be any sequence of sets. An element of $X \otimes Y$ is a plane tree plane tree $T$, where every non-leaf has at least $2$ children, where every leaf is labeled with an element of $X \sqcup Y$, and every non-leaf with $i$ children is labeled with an element of $B_i$, subject to the condition:

If $v$ is a non-leaf but all the children of $v$ are leaves, then the children of $v$ do not come entirely from $X$ nor entirely from $Y$.

To see associativity, note that both $(X \otimes Y) \otimes Z$ and $X \otimes (Y \otimes Z)$ can be canonically bisected with plane trees whose non-leaves are labeled with $X \sqcup Y \sqcup Z$, subject to the analogous conditions. (The analogue of the boxed condition is that the children of $v$ do not come entirely from $X$, nor entirely from $Y$, nor entirely from $Z$.)

I think the logarithm of the corresponding group law is $t - \sum_{i \geq 2} |B_i| t^i$. Since this method can only categorify group laws whose logarithm has integer coefficients, this clearly cannot include many of the most obvious examples.

The following family of examples can be extracted from Lenart and Ray, Some applications of incidence Hopf algebras to formal group theory and algebraic topology (Postscript). Let $B_2$, $B_3$, $B_4$, … be any sequence of sets. An element of $X \otimes Y$ is a plane tree $T$, where every non-leaf has at least $2$ children, where every leaf is labeled with an element of $X \sqcup Y$, and every non-leaf with $i$ children is labeled with an element of $B_i$, subject to the condition:

If $v$ is a non-leaf but all the children of $v$ are leaves, then the children of $v$ do not come entirely from $X$ nor entirely from $Y$.

To see associativity, note that both $(X \otimes Y) \otimes Z$ and $X \otimes (Y \otimes Z)$ can be canonically bisected with plane trees whose non-leaves are labeled with $X \sqcup Y \sqcup Z$, subject to the analogous conditions. (The analogue of the boxed condition is that the children of $v$ do not come entirely from $X$, nor entirely from $Y$, nor entirely from $Z$.)

I think the logarithm of the corresponding group law is $t - \sum_{i \geq 2} |B_i| t^i$. Since this method can only categorify group laws whose logarithm has integer coefficients, this clearly cannot include many of the most obvious examples.

The following family of examples can be extracted from Lenart and Ray, Some applications of incidence Hopf algebras to formal group theory and algebraic topology (Postscript). Let $B_2$, $B_3$, $B_4$, … be any sequence of sets. An element of $X \otimes Y$ is a plane tree $T$, where every non-leaf has at least $2$ children, where every leaf is labeled with an element of $X \sqcup Y$, and every non-leaf with $i$ children is labeled with an element of $B_i$, subject to the condition:

If $v$ is a non-leaf but all the children of $v$ are leaves, then the children of $v$ do not come entirely from $X$ nor entirely from $Y$.

To see associativity, note that both $(X \otimes Y) \otimes Z$ and $X \otimes (Y \otimes Z)$ can be canonically bisected with plane trees whose non-leaves are labeled with $X \sqcup Y \sqcup Z$, subject to the analogous conditions. (The analogue of the boxed condition is that the children of $v$ do not come entirely from $X$, nor entirely from $Y$, nor entirely from $Z$.)

I think the logarithm of the corresponding group law is $t - \sum_{i \geq 2} |B_i| t^i$. Since this method can only categorify group laws whose logarithm has integer coefficients, this clearly cannot include many of the most obvious examples.

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created Jan 29, 2014 at 2:31

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The following family of examples can be extracted from Lenart and Ray, Some applications of incidence Hopf algebras to formal group theory and algebraic topology (Postscript). Let $B_2$, $B_3$, $B_4$, … be any sequence of sets. An element of $X \otimes Y$ is a plane tree $T$, where every non-leaf has at least $2$ children, where every leaf is labeled with an element of $X \sqcup Y$, and every non-leaf with $i$ children is labeled with an element of $B_i$, subject to the condition:

If $v$ is a non-leaf but all the children of $v$ are leaves, then the children of $v$ do not come entirely from $X$ nor entirely from $Y$.

To see associativity, note that both $(X \otimes Y) \otimes Z$ and $X \otimes (Y \otimes Z)$ can be canonically bisected with plane trees whose non-leaves are labeled with $X \sqcup Y \sqcup Z$, subject to the analogous conditions. (The analogue of the boxed condition is that the children of $v$ do not come entirely from $X$, nor entirely from $Y$, nor entirely from $Z$.)

I think the logarithm of the corresponding group law is $t - \sum_{i \geq 2} |B_i| t^i$. Since this method can only categorify group laws whose logarithm has integer coefficients, this clearly cannot include many of the most obvious examples.