Are valued-quiver species and combinatorial species related?
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In these slides, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i in Q_0$ we assign a division ring $mathbf{k}_i$, and to each arrow $(a colon i to j) in Q_1$ we assign a $(mathbf{k}_j, mathbf{k}_i)$-bimodule $M_a$ that satisfies a couple of conditions:
$operatorname{Hom}_{mathbf{k}_j}(M_a, mathbf{k}_j) simeq operatorname{Hom}_{mathbf{k}_i}(M_a, mathbf{k}_i)$ as $(mathbf{k}_i, mathbf{k}_j)$-bimodules.
$(dim_{mathbf{k}_j}M_a, dim_{mathbf{k}_i}M_a)$ is equal to the pair of values on the edge $a$.
Now a combinatorial species is an endofunctor of the category of finite sets, where the morphisms are set bijections. Is this first sort of species related at all to a combinatorial species? or do these two types
of objects just coincidentally (unfortunately?) have the same name? And if they're not the same, what is the origin of the valued-quiver species? The term for combinatorial species comes from Andre Joyal's "espèces de structure".
category-theory representation-theory quiver combinatorial-species
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
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add a comment |
$begingroup$
In these slides, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i in Q_0$ we assign a division ring $mathbf{k}_i$, and to each arrow $(a colon i to j) in Q_1$ we assign a $(mathbf{k}_j, mathbf{k}_i)$-bimodule $M_a$ that satisfies a couple of conditions:
$operatorname{Hom}_{mathbf{k}_j}(M_a, mathbf{k}_j) simeq operatorname{Hom}_{mathbf{k}_i}(M_a, mathbf{k}_i)$ as $(mathbf{k}_i, mathbf{k}_j)$-bimodules.
$(dim_{mathbf{k}_j}M_a, dim_{mathbf{k}_i}M_a)$ is equal to the pair of values on the edge $a$.
Now a combinatorial species is an endofunctor of the category of finite sets, where the morphisms are set bijections. Is this first sort of species related at all to a combinatorial species? or do these two types
of objects just coincidentally (unfortunately?) have the same name? And if they're not the same, what is the origin of the valued-quiver species? The term for combinatorial species comes from Andre Joyal's "espèces de structure".
category-theory representation-theory quiver combinatorial-species
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
$endgroup$
5
$begingroup$
I have the very strong impression that it is just a coincidence, but I would love to be proven wrong!
$endgroup$
– Ivan Di Liberti
Dec 3 '18 at 23:40
add a comment |
$begingroup$
In these slides, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i in Q_0$ we assign a division ring $mathbf{k}_i$, and to each arrow $(a colon i to j) in Q_1$ we assign a $(mathbf{k}_j, mathbf{k}_i)$-bimodule $M_a$ that satisfies a couple of conditions:
$operatorname{Hom}_{mathbf{k}_j}(M_a, mathbf{k}_j) simeq operatorname{Hom}_{mathbf{k}_i}(M_a, mathbf{k}_i)$ as $(mathbf{k}_i, mathbf{k}_j)$-bimodules.
$(dim_{mathbf{k}_j}M_a, dim_{mathbf{k}_i}M_a)$ is equal to the pair of values on the edge $a$.
Now a combinatorial species is an endofunctor of the category of finite sets, where the morphisms are set bijections. Is this first sort of species related at all to a combinatorial species? or do these two types
of objects just coincidentally (unfortunately?) have the same name? And if they're not the same, what is the origin of the valued-quiver species? The term for combinatorial species comes from Andre Joyal's "espèces de structure".
category-theory representation-theory quiver combinatorial-species
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
$endgroup$
In these slides, Joel Lemay defines a species as a generalization of a valued quiver $Q$, where to each node $i in Q_0$ we assign a division ring $mathbf{k}_i$, and to each arrow $(a colon i to j) in Q_1$ we assign a $(mathbf{k}_j, mathbf{k}_i)$-bimodule $M_a$ that satisfies a couple of conditions:
$operatorname{Hom}_{mathbf{k}_j}(M_a, mathbf{k}_j) simeq operatorname{Hom}_{mathbf{k}_i}(M_a, mathbf{k}_i)$ as $(mathbf{k}_i, mathbf{k}_j)$-bimodules.
$(dim_{mathbf{k}_j}M_a, dim_{mathbf{k}_i}M_a)$ is equal to the pair of values on the edge $a$.
Now a combinatorial species is an endofunctor of the category of finite sets, where the morphisms are set bijections. Is this first sort of species related at all to a combinatorial species? or do these two types
of objects just coincidentally (unfortunately?) have the same name? And if they're not the same, what is the origin of the valued-quiver species? The term for combinatorial species comes from Andre Joyal's "espèces de structure".
category-theory representation-theory quiver combinatorial-species
category-theory representation-theory quiver combinatorial-species
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
edited Dec 11 '18 at 4:14
Mike Pierce
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
asked Dec 3 '18 at 23:23
Mike PierceMike Pierce
11.4k103584
11.4k103584
5
$begingroup$
I have the very strong impression that it is just a coincidence, but I would love to be proven wrong!
$endgroup$
– Ivan Di Liberti
Dec 3 '18 at 23:40
add a comment |
5
$begingroup$
I have the very strong impression that it is just a coincidence, but I would love to be proven wrong!
$endgroup$
– Ivan Di Liberti
Dec 3 '18 at 23:40
5
5
$begingroup$
I have the very strong impression that it is just a coincidence, but I would love to be proven wrong!
$endgroup$
– Ivan Di Liberti
Dec 3 '18 at 23:40
$begingroup$
I have the very strong impression that it is just a coincidence, but I would love to be proven wrong!
$endgroup$
– Ivan Di Liberti
Dec 3 '18 at 23:40
add a comment |
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I have the very strong impression that it is just a coincidence, but I would love to be proven wrong!
$endgroup$
– Ivan Di Liberti
Dec 3 '18 at 23:40