Lallement functor is a weak right multiadjoint
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915688848818176 |
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| author | Vidal, Juan Climent Llópez, Enric Cosme |
| author_facet | Vidal, Juan Climent Llópez, Enric Cosme |
| contents | For a plural signature $Σ$ and with regard to the category $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$, of naturally preordered idempotent $Σ$-algebras and surjective homomorphisms, we define a contravariant functor $\mathrm{Lsys}_Σ$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, the category of categories, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $\mathbf{I}$-$\mathsf{LAlg}(Σ)$, of $\mathbf{I}$-semi-inductive Lallement systems of $Σ$-algebras, and a covariant functor $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \mathbf{I})$, of the coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which $\mathbf{A}$ is a $Σ$-algebra and $f\colon \mathbf{A}\longrightarrow \mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories $\int^{\mathsf{NPIAlg}(Σ)_{\mathsf{s}}}\mathrm{Lsys}_Σ$ and $\int_{\mathsf{NPIAlg}(Σ)_{\mathsf{s}}}(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$; define a functor $\mathfrak{L}_Σ$ from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_02944 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lallement functor is a weak right multiadjoint Vidal, Juan Climent Llópez, Enric Cosme Category Theory 08A62, 08B25, 08C05, 18A40 For a plural signature $Σ$ and with regard to the category $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$, of naturally preordered idempotent $Σ$-algebras and surjective homomorphisms, we define a contravariant functor $\mathrm{Lsys}_Σ$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, the category of categories, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $\mathbf{I}$-$\mathsf{LAlg}(Σ)$, of $\mathbf{I}$-semi-inductive Lallement systems of $Σ$-algebras, and a covariant functor $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \mathbf{I})$, of the coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which $\mathbf{A}$ is a $Σ$-algebra and $f\colon \mathbf{A}\longrightarrow \mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories $\int^{\mathsf{NPIAlg}(Σ)_{\mathsf{s}}}\mathrm{Lsys}_Σ$ and $\int_{\mathsf{NPIAlg}(Σ)_{\mathsf{s}}}(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$; define a functor $\mathfrak{L}_Σ$ from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor. |
| title | Lallement functor is a weak right multiadjoint |
| topic | Category Theory 08A62, 08B25, 08C05, 18A40 |
| url | https://arxiv.org/abs/2311.02944 |