Wedge Products and Cotensor Coalgebras in Monoidal Categories
Wedge Products and Cotensor Coalgebras in Monoidal Categories
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DOI：
10.1017/is007011015jkt012
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发表时间：
2004-07
影响因子：
0.6
通讯作者：
A. Ardizzoni
中科院分区：
文献类型：
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作者：
A. Ardizzoni
Ardizzoni 等人构建了“阿贝尔幺半群”范畴 $\mathcal{M}$ 的余张量余代数，该范畴也是余完备、完备和 AB5。 （通信代数 35(1):25–70, 2007）。还证明了该余代数满足类似于经典的有意义的普遍性质。这里，通过考虑由 E 的子代数 D 的楔积组成的过滤的直接极限 $\widetilde{D}$ 来填补 $\mathcal{M}$ 中余代数 E 的共代数过滤的缺失。本文的主要目的是通过余张量来表征遗传余代数 $\widetilde{D}$，其中 D 是 $\mathcal{M}$ 中的可分余代数煤代数：更准确地说，我们证明，在适当的假设下，$\widetilde{D}$ 是遗传的当且仅当它是形式光滑的当且仅当它是余张余代数 $T^c_{D}(D\wedge_E D/D)$ 当且仅当它是余张余余代数 $T^c_{D}(N)$，其中 N 是 $\mathcal{M}$ 中的某个 D-bico 模。由于我们的选择，即使我们将结果应用到向量空间类别中，也会获得新的结果。
The construction of the cotensor coalgebra for an “abelian monoidal” category $\mathcal{M}$ which is also cocomplete, complete and AB5, was performed in Ardizzoni et al. (Comm Algebra 35(1):25–70, 2007). It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in $\mathcal{M}$ is filled by considering a direct limit $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras $\widetilde{D}$, where D is a coseparable coalgebra in $\mathcal{M}$, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, $\widetilde{D}$ is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra $T^c_{D}(D\wedge_E D/D)$ if and only if it is a cotensor coalgebra $T^c_{D}(N)$, where N is a certain D-bicomodule in $\mathcal{M}$. Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained.