拓扑中的Lawvere-Tierney层、因子分解系统、截面和$j$ -本质单同态
Lawvere-Tierney sheaves, factorization systems, sections and $j$-essential monomorphisms in a topos
摘要：
设$j$是任意拓扑$\mathcal{E}$上的Lawvere-Tierney拓扑（简称拓扑），$B$是$\mathcal{E}$的对象，而$j_B = j\times 1_B$是切片拓扑$\mathcal{E}/B$上的诱导拓扑。本文分析了拉回函子$Π_B:\mathcal{E}\rightarrow \mathcal{E}/B$与拓扑有关的一些性质。然后对拓扑$\mathcal{E}$中所有$j$ -稠密单态的左可消类$\mathcal{M}$，得到了$(\mathcal{M} , \mathcal{M}^{\perp})$是$\mathcal{E}$中的因式分解系统的一些充要条件，这与切片拓扑$\mathcal{E}/B,$中的因式分解系统有关，其中$B$在$\mathcal{E}$的对象类上的范围内.除其他事项外，我们证明了箭头$f : X\rightarrow B$在$\mathcal{E}$是一个$j_B$ -层每当图的$f$，是一个部分在$\mathcal{E}/B$以及对象的部分$S(f)$的$f$，是一个$j$ -层在$\mathcal{E}$。此外，我们在$\mathcal{E}$中引入了一类单态，我们称之为$j$ -必需的。给出了它们的一些等价形式及其性质。同时，我们证明了在一个预层拓扑中的任何预层都有一个极大本质扩张。最后，讨论了用一些作者研究的（产生性）弱拓扑$j$代替拓扑时所得结果的异同。
英文摘要：
Let $j$ be a Lawvere-Tierney topology (a topology, for short) on an arbitrary topos $\mathcal{E}$, $B$ an object of $\mathcal{E}$, and $j_B = j\times 1_B$ the induced topology on the slice topos $\mathcal{E}/B$. In this manuscript, we analyze some properties of the pullback functor $Π_B:\mathcal{E}\rightarrow \mathcal{E}/B$ which have deal with topology. Then for a left cancelable class $\mathcal{M}$ of all $j$-dense monomorphisms in a topos $\mathcal{E}$, we achieve some necessary and sufficient conditions for that $(\mathcal{M} , \mathcal{M}^{\perp})$ is a factorization system in $\mathcal{E}$, which is related to the factorization systems in slice topoi $\mathcal{E}/B,$ where $B$ ranges over the class of objects of $\mathcal{E}$. Among other things, we prove that an arrow $f : X\rightarrow B$ in $\mathcal{E}$ is a $j_B$-sheaf whenever the graph of $f$, is a section in $\mathcal{E}/B$ as well as the object of sections $S(f)$ of $f$, is a $j$-sheaf in $\mathcal{E}$. Furthermore, we introduce a class of monomorphisms in $\mathcal{E}$, which we call them $j$-essential. Some equivalent forms of those and some of their properties are presented. Also, we prove that any presheaf in a presheaf topos has a maximal essential extension. Finally, some similarities and differences of the obtained result are discussed if we put a (productive) weak topology $j$, studied by some authors, instead of a topology.
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