Complete W*-categories
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910682426900480 |
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| author | Henriques, André Nivedita Penneys, David |
| author_facet | Henriques, André Nivedita Penneys, David |
| contents | We study $\mathrm{W}^*$-categories, and explain the ways in which complete $\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\langle\,\,,\,\rangle_{\mathrm{Hilb}}\,:\,\overline C\times C\,\to\, \mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\mathrm{W}^*$-categories there is an antilinear equivalence $$\dagger:\mathrm{Func}(C,D) \leftrightarrow \mathrm{Func}(D,C)$$ characterised by $\langle c,F^\dagger(d)\rangle_{\mathrm{Hilb}} \simeq \langle F(c),d\rangle_{\mathrm{Hilb}}$, for $c\in C$ and $d \in D$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complete W*-categories Henriques, André Nivedita Penneys, David Operator Algebras Category Theory We study $\mathrm{W}^*$-categories, and explain the ways in which complete $\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\langle\,\,,\,\rangle_{\mathrm{Hilb}}\,:\,\overline C\times C\,\to\, \mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\mathrm{W}^*$-categories there is an antilinear equivalence $$\dagger:\mathrm{Func}(C,D) \leftrightarrow \mathrm{Func}(D,C)$$ characterised by $\langle c,F^\dagger(d)\rangle_{\mathrm{Hilb}} \simeq \langle F(c),d\rangle_{\mathrm{Hilb}}$, for $c\in C$ and $d \in D$. |
| title | Complete W*-categories |
| topic | Operator Algebras Category Theory |
| url | https://arxiv.org/abs/2411.01678 |