Title: The higher algebra of weighted colimits Show Chinese title

Title: 加权余极限的高阶代数

Abstract: We develop a theory of weighted colimits in the framework of weakly bienriched $\infty$-categories, an extension of Lurie's notion of enriched $\infty$-categories. We prove an existence result for weighted colimits, study weighted colimits of diagrams of enriched functors, express weighted colimits via enriched coends, characterize the enriched $\infty$-category of enriched presheaves as the free cocompletion under weighted colimits, prove a Bousfield-Kan formula for weighted colimits and an enriched adjoint functor theorem and develop a theory of universally adjoining weighted colimits to an enriched $\infty$-category. Via the latter we construct for every presentably $\mathbb{E}_{k+1}$-monoidal $\infty$-category $\mathcal{V}$ for $1 \leq k \leq \infty$ and set $\mathcal{H}$ of weights a presentably $\mathbb{E}_k$-monoidal structure on the $\infty$-category of $\mathcal{V}$-enriched $\infty$-categories that admit $\mathcal{H}$-weighted colimits. Varying $\mathcal{H}$ this $\mathbb{E}_k$-monoidal structure interpolates between the tensor product for $\mathcal{V}$-enriched $\infty$-categories and the relative tensor product for $\infty$-categories presentably left tensored over $\mathcal{V}$. Studying functoriality in $\mathcal{H}$ we deduce that taking $\mathcal{V}$-enriched presheaves is $\mathbb{E}_k$-monoidal with respect to the tensor product on small $\mathcal{V}$-enriched $\infty$-categories and the relative tensor product on $\infty$-categories presentably left tensored over $\mathcal{V}.$ As key applications we construct for every $n \geq 1 $ and set $\mathcal{K}$ of $(\infty, n)$-categories a tensor product for $(\infty,n)$-categories that admit $\mathcal{K}$-indexed (op)lax colimits, a tensor product for Cauchy-complete $\mathcal{V}$-enriched $\infty$-categories and tensor products for (Cauchy complete) $n$-stable, $n$-additive and $n$-preadditive $(\infty,n)$-categories. Show Chinese abstract

Abstract: 我们在弱双丰富$\infty$-范畴的框架下发展了加权极限的理论，这是对Lurie的丰富$\infty$-范畴概念的扩展。我们证明了加权极限的存在性结果，研究了丰富函子图的加权极限，通过丰富协边积分表达了加权极限，将丰富$\infty$-范畴的丰富预层表为在加权极限下的自由完备化，证明了加权极限的Bousfield-Kan公式和一个丰富的伴随函子定理，并发展了一个将加权极限普遍地附加到丰富$\infty$-范畴的理论。 通过后者，我们为每个可表示的$\mathbb{E}_{k+1}$-单oidal$\infty$-范畴$\mathcal{V}$对于$1 \leq k \leq \infty$和设置$\mathcal{H}$的权重，在$\infty$-范畴的$\mathcal{V}$-丰富$\infty$-范畴上构造一个可表示的$\mathbb{E}_k$-单oidal 结构，这些结构接受$\mathcal{H}$-加权余极限。 变化 $\mathcal{H}$这个$\mathbb{E}_k$-单子结构在$\mathcal{V}$-丰富$\infty$-范畴的张量积和在$\infty$-范畴中相对于$\mathcal{V}$的相对张量积之间进行插值。 研究$\mathcal{H}$的函子性，我们得出结论，取$\mathcal{V}$-丰富预层在小$\mathcal{V}$-丰富$\infty$-范畴上的张量积以及在$\infty$-范畴上相对于$\mathcal{V}.$的相对张量积时，是$\mathbb{E}_k$-单性的。作为关键应用，我们为每个$n \geq 1 $和集合$\mathcal{K}$的$(\infty, n)$-范畴构造了一个张量积，该张量积适用于具有$\mathcal{K}$-索引（对）弱余极限的$(\infty,n)$-范畴，适用于柯西完备的$\mathcal{V}$-丰富$\infty$-范畴的张量积，以及适用于（柯西完备）$n$-稳定、$n$-加性和$n$-预加性$(\infty,n)$-范畴的张量积。