自由生成的n-范畴、上插入器与低维范畴的表示
Freely generated $n$-categories, coinserters and presentations of low dimensional categories
AI总结：
研究自由生成n-范畴等相关问题，核心方法是通过上插入器和上均衡器给出计算表示的统一描述，主要贡献为从计算体恢复自由严格n-范畴，满足双范畴泛性质，给出群胚亏格等相关结论及同调下界等。
摘要：
一个表示不仅记录了一个范畴结构，还记录了它是如何组装的。这对于重写、一致性和极小性很重要：自由地添加一个具有规定边界的胞腔与在已构建的胞腔之间施加一个等式是不同的。我们表明这两个操作分别由上插入器和上均衡器控制，从普通范畴到严格高阶范畴给出了计算表示的统一描述。对于一个图G，G上的自由范畴是其离散范畴之间的定义域和余定义域映射在范畴中的上插入器。更一般地，对于每个n≥1，自由地将具有规定平行边界的n-胞腔添加到一个严格的(n - 1)-范畴是严格n-范畴、严格n-函子和n-图标2-范畴中的上插入器。此构造是底层推导方案函子的左伴随。它从计算体恢复自由严格n-范畴，而上均衡器在自由生成的胞腔之间施加等式。在二维中，它还满足关于正规伪函子和图标的双范畴泛性质。用单位区间代替行走箭头得到图的拓扑上插入器。对于一个群胚2-计算体C，为每个关系附加一个圆盘产生一个表示复形XC，其基本群胚是由C表示的群胚。因此，一个连通群胚的秩有限亏格是任何各向同性群的经典亏格。同调给出了在固定图上表示一个薄群胚所需关系的尖锐下界，而交叉模将比较扩展到关系之间的关系。最后，对于下降计算体，恒等和结合律合流形成一个同伦基并达到相应的同调下界。
英文摘要：
A presentation records not only a categorical structure but also how it is assembled. This matters for rewriting, coherence, and minimality: freely adjoining a cell with prescribed boundary is different from imposing an equation between cells already constructed. We show that these two operations are governed, respectively, by coinserters and coequifiers, giving a uniform account of computadic presentations from ordinary categories to strict higher categories. For a graph $G$, the free category on $G$ is the coinserter in $\mathsf{Cat}$ of its domain and codomain maps between discrete categories. More generally, for every $n\geq1$, freely adjoining $n$-cells with prescribed parallel boundaries to a strict $(n-1)$-category is a coinserter in the $2$-category of strict $n$-categories, strict $n$-functors, and $n$-icons. This construction is left adjoint to the underlying derivation-scheme functor. It recovers free strict $n$-categories from computads, while coequifiers impose equations between freely generated cells. In dimension two, it also satisfies a bicategorical universal property for normal pseudofunctors and icons. Replacing the walking arrow by the unit interval yields the topological coinserter of a graph. For a groupoidal $2$-computad $C$, attaching one disk for each relation produces a presentation complex $X_C$ whose fundamental groupoid is the groupoid presented by $C$. Consequently, the rank-finite deficiency of a connected groupoid is the classical deficiency of any isotropy group. Homology gives a sharp lower bound on the relations required to present a thin groupoid over a fixed graph, while crossed modules extend the comparison to relations among relations. Finally, for the descent computad, the identity and associativity confluences form a homotopy basis and attain the corresponding homological lower bounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
