Hopf substitutions in Species
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908957653598208 |
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| author | Lauve, Aaron Lazzeroni, Anthony |
| author_facet | Lauve, Aaron Lazzeroni, Anthony |
| contents | In the theory of species, the species $\mathbf{L}$ of linear orders and the substitution operation $\boldsymbol{\circ}$ combine for a compelling result: given any positive comonoid $\mathbf{p}$, $\mathbf{L}\boldsymbol{\circ}\mathbf{p}$ carries the structure of Hopf monoid, freely generated by $\mathbf{p}$. Leaving aside the universal property this implies, we ask, "for which $\mathbf{b}$ does $\mathbf{b}\boldsymbol{\circ}\mathbf{p}$ carry the structure of Hopf monoid?" After answering this question, we look at basic properties of our construction. We also extend a result of the present authors, on interpolation in species, to this new context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10816 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hopf substitutions in Species Lauve, Aaron Lazzeroni, Anthony Combinatorics Category Theory 18M80, 16T30 In the theory of species, the species $\mathbf{L}$ of linear orders and the substitution operation $\boldsymbol{\circ}$ combine for a compelling result: given any positive comonoid $\mathbf{p}$, $\mathbf{L}\boldsymbol{\circ}\mathbf{p}$ carries the structure of Hopf monoid, freely generated by $\mathbf{p}$. Leaving aside the universal property this implies, we ask, "for which $\mathbf{b}$ does $\mathbf{b}\boldsymbol{\circ}\mathbf{p}$ carry the structure of Hopf monoid?" After answering this question, we look at basic properties of our construction. We also extend a result of the present authors, on interpolation in species, to this new context. |
| title | Hopf substitutions in Species |
| topic | Combinatorics Category Theory 18M80, 16T30 |
| url | https://arxiv.org/abs/2604.10816 |