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Weighted limits in an $$(\infty ,1)$$ ( ∞ , 1 ) -category
Applied Categorical Structures ( IF 0.6 ) Pub Date : 2021-05-05 , DOI: 10.1007/s10485-021-09643-z
Martina Rovelli

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.



中文翻译:

$$(\ infty,1)$$(∞,1)-类中的加权限制

我们在任意准类别中引入加权极限的概念,在一般类别中适当地推广普通极限,而在普通类别中引入经典加权极限。这是通过概括Joyal的方法来完成的:我们在拟类别中的图上确定加权锥的拟类别的有意义的构造,该类的最终对象是所考虑图的加权极限。然后,我们证明每个加权极限可以表示为普通极限。当准类别作为富集在Kan络合物上的类别的同伦同调相干神经出现时,我们概括了Riehl-Verity的论点,以证明加权极限与富集类别理论意义上的同伦加权极限一致,对此,显式结构是可利用的。准类别完成后,

更新日期:2021-05-05
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