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Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:

Definition. A simplicial set $K$ is $\kappa$-sifted if, for every set $E$ with $\lvert E\rvert<\kappa$, the diagonal functor $K\to K^E$ is cofinal.

It follows from [Lurie: Higher Topos Theory, Prop 5.3.1.22] that every $\kappa$-filtered $\infty$-category is $\kappa$-sifted. On the other hand, the category ${\mathbf\Delta}^{\operatorname{op}}$ seems to be $\kappa$-sifted since the product of any $\kappa$'s simplices is still weakly contractible.

I wonder:

  1. Is there any study of this notion in the literature?

  2. To what extent, $\kappa$-sifted colimits = $\kappa$-filtered colimits + geometric realizations? For example, let $F\colon\mathcal C\to\mathcal D$ be a functor. Under what mild assumptions on $\mathcal C$ and $\mathcal D$, it is true that, if $F$ preserves geometric realizations and is $\kappa$-accessible, then it preserves $\kappa$-sifted colimits? I am aware of the following references when $\kappa=\aleph_0$:

    1. Suppose that $\mathcal C$ has small colimits. Then it is true, as proved in [Joyal: On Logoi, §33.24] or [Lurie: Higher Topos Theory, Cor 5.5.8.17].

    2. In the 1-category situation, and suppose that $\mathcal C$ has small limits, it is true, as proved in [Adámek–Rosický–Vitale, On Algebraically Exact Categories and Essential Localizations of Varieties].

    However, I am more interested in the situation that $\kappa>\aleph_0$, and hopefully it is simpler than the case that $\kappa=\aleph_0$.

Update: it seems that usual constructions and statements work for $\kappa$-sifted case as well. For example, let $\mathcal C^0$ be a small $\infty$-category with $\kappa$-small coproducts. Then we can define $\mathcal P_{\Sigma^\kappa}$ to be the $\infty$-category of $\kappa$-small-product-preserving presheaves $(\mathcal C^0)^{\operatorname{op}}\to\operatorname{An}$, and in this situation, letting $\mathcal D$ be a cocomplete $\infty$-category, then $\kappa$-sifted-colimit-preserving functors $\mathcal P_{\Sigma^\kappa}(\mathcal C^0)\to\mathcal D$ is the same as functors $\mathcal C^0\to\mathcal D$, and so forth.

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    $\begingroup$ Homotopy colimits over Δ^op do not preserve infinite homotopy products (e.g., take the product of countably many copies of the nerve of N(Z,<)), so Δ^op does not satisfy the claimed property. $\endgroup$
    – Dmitri Pavlov
    Commented Jun 30 at 14:01
  • $\begingroup$ @DmitriPavlov Thanks. I am confused what is happening: say, $([m_n])_n\in({\mathbf\Delta}^{\operatorname{op}})^{\mathbb N}$, then Quillen's Theorem A reduces to check the weak contractibility of $\prod_n\Delta^{m_n}$. What's wrong in this argument? $\endgroup$
    – Z. M
    Commented Jun 30 at 14:20
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    $\begingroup$ How do you intend to relate the diagonal functor to the condition of commuting sifted colimits and infinite products? For commuting sifted colimits and finite products, such a relationship is established by a (finite) induction on the number of factors, using the fact that products with a fixed object preserve colimits. $\endgroup$
    – Dmitri Pavlov
    Commented Jun 30 at 16:10
  • $\begingroup$ Just in case you are interested: This question is related to mathoverflow.net/questions/453235/… $\endgroup$
    – Georg Lehner
    Commented Jul 1 at 17:11

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In the one-dimensional setting, $\kappa$-sifted categories are studied in §3 of Adámek–Koubek–Velebil's A duality between infinitary varieties and algebraic theories. However, it is shown there (Theorem 3.1) that the $\kappa$-sifted categories coincide with the $\kappa$-filtered categories for $\kappa$ uncountable; thus the notion is uninteresting except when $\kappa = \aleph_0$.

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  • $\begingroup$ It seems that this definition is different from mine. Fix an uncountable regular cardinal $\kappa$. Let $R$ be a ring with a monogenic non-flat $R$-module $M=R/r$. Then the category $\mathcal C$ of free $R$-modules $F$ of rank $<\kappa$ equipped with a map $F\to M$ of $R$-modules has $\kappa$-small coproducts, thus it is $\kappa$-sifted in my definition. However, it does not seem to be filtered, let alone being $\kappa$-filtered: the colimit of the functor $\mathcal C\to\operatorname{Mod}_R,N\mapsto N$ seems to be $M$, but by Lazard's theorem, if $\mathcal C$ is filtered, then $M$ is flat. $\endgroup$
    – Z. M
    Commented Jun 30 at 20:38
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    $\begingroup$ The real subtlety is that the doctrine of $\kappa$-small sets is probably not sound, that is to say, filtered and weakly filtered categories with respect to this doctrine might not coincide (I am referring the terminology in Adámek–Borceaux–Lack–Rosický and its $\infty$-categorical analogue explained in Rezk's draft). My definition corresponds to being weakly filtered. $\endgroup$
    – Z. M
    Commented Jul 1 at 9:41
  • $\begingroup$ For future readers: It was established on discord that there are two working definitions of $\kappa$-sifted: - Either the weak version, as defined by @Z. M above in terms of the diagonal functors being cofinal. - The strong version, as used by Adámek et al. defining $C$ to be $\kappa$-sifted if $C$-indexed colimits in Sets (1-categorically) or Spaces ($\infty$-categorically) commute with $\kappa$-small products. The weak version does not need to imply the strong version - This is where the confusion comes from. $\endgroup$
    – Georg Lehner
    Commented Jul 1 at 17:09

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