On logical parameterizations and functional representability in local set theories
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911776609665024 |
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| author | Hernández, Enrique Ruiz Solórzano, Pedro |
| author_facet | Hernández, Enrique Ruiz Solórzano, Pedro |
| contents | There is a well-known inclusion $ι_\mathscr{E}$ of a topos $\mathscr{E}$ in the linguistic topos $\mathscr{T}(Σ)$ of its internal language $Σ$ that proves both toposes to be equivalent. There is also a canonical translation $η_S$ for any local set theory $S$ into the local set theory $Σ$ of its linguistic topos. Starting from a local set theory, this yields two a priori distinct inclusions from $\mathscr{T}(S)$ to $\mathscr{T}(Σ)$.
Herein, these two functors are proved to be isomorphic. Furthermore, the concept of logical parameterization is investigated and then applied to see that $ι_{\mathscr{T}(S)}$ parameterizes $\mathscr{T}(η_S)$ in such a way that syntactic $S$-functions are represented by themselves in $Σ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_07405 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On logical parameterizations and functional representability in local set theories Hernández, Enrique Ruiz Solórzano, Pedro Logic Category Theory Primary 18C50, Secondary 03B38, 03F50 There is a well-known inclusion $ι_\mathscr{E}$ of a topos $\mathscr{E}$ in the linguistic topos $\mathscr{T}(Σ)$ of its internal language $Σ$ that proves both toposes to be equivalent. There is also a canonical translation $η_S$ for any local set theory $S$ into the local set theory $Σ$ of its linguistic topos. Starting from a local set theory, this yields two a priori distinct inclusions from $\mathscr{T}(S)$ to $\mathscr{T}(Σ)$. Herein, these two functors are proved to be isomorphic. Furthermore, the concept of logical parameterization is investigated and then applied to see that $ι_{\mathscr{T}(S)}$ parameterizes $\mathscr{T}(η_S)$ in such a way that syntactic $S$-functions are represented by themselves in $Σ$. |
| title | On logical parameterizations and functional representability in local set theories |
| topic | Logic Category Theory Primary 18C50, Secondary 03B38, 03F50 |
| url | https://arxiv.org/abs/2104.07405 |