A toolbox for weighted $(\mathrm{\infty },n)$-limits

An $(\mathrm{\infty },n)$-category consists of objects and an $(\mathrm{\infty },n-1)$-category of higher morphisms between any two objects, while an $(\mathrm{\infty },0)$-category is simply a space. In this setting, many constructions of interest have the universal property of the limit of an $(\mathrm{\infty },n)$-functor $F:𝒥\to 𝒞$, possibly weighted by an $(\mathrm{\infty },n)$-functor $W:𝒥\to 𝒞a{t}_{(\mathrm{\infty },n-1)}$.

Developing a solid theory of limits and colimits in the $(\mathrm{\infty },n)$-categorical setting for $n>0$ has been a central goal of my research program with Moser and Rasekh. In earlier work [1, 2], we proposed and validated a definition of weighted limits and colimits in this setting, and in this talk we will report on current developments of the theory [3].

We will primarily discuss completeness results and provide explicit formulas for weighted (co)limits under specific assumptions on $𝒞$. These include:

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  a formula for weighted limits in $𝒞$ in terms of products, (co)tensors, limits over the simplex category, under the assumption that $𝒞$ admits these elementary (co)limits;

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  a formula for weighted limits in $𝒞$ in terms of the Grothendieck constructions ${\int }_{𝒥}F$ and ${\int }_{𝒥}W$ of $F$ and $W$, when $𝒞$ is $𝒞a{t}_{(\mathrm{\infty },n-1)}$;

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  a pointwise formula for weighted limits in $𝒞$, when $𝒞$ is an $(\mathrm{\infty },n)$-category of functors valued in a suitably complete $(\mathrm{\infty },n)$-category.

If time permits, we will also indicate how several classical theorems on limits and colimits extend to the $(\mathrm{\infty },n)$-categorical setting. These include:

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  a generalization of Fubini’s formula, describing the commutation of weighted $(\mathrm{\infty },n)$-limits with weighted $(\mathrm{\infty },n)$-limits;

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  a characterization of weighted cofinality for $(\mathrm{\infty },n)$-functors;

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  a cancellation property for comma objects in an $(\mathrm{\infty },n)$-category; and

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  a construction of the free completion of an $(\mathrm{\infty },n)$-category under weighted limits.