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LSpice
LSpice
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$Ind $\operatorname{Ind}(C^I) = Ind\operatorname{Ind}(C)^I$?

Let$\DeclareMathOperator\Ind{Ind}$Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $Ind(C^I) \to Ind(C)^I$$\Ind(C^I) \to \Ind(C)^I$ is an equivalence of categories? where Ind$\Ind$ denotes the category of ind-objects ( soso the free completion under filtered colimits) and the exponentialexponentials are for categories of functors.

This is proved Byby Lurie in Higher topos theory (proposition 5.3.5.15) when $C$ is an infinity category and $I$ is a finite poset. He gives an example to show that this cannot be generalized to the case of $I$ a finite simplicial sets -set — but finite category is much more restrictive and his example doesn't rule this out at all.

I'm mostly interested in the case where $C$ is a 1-category and already has finite colimits, but I'm curious of any interesting things that can be said more generally.

$Ind(C^I) = Ind(C)^I$?

Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $Ind(C^I) \to Ind(C)^I$ is an equivalence of categories? where Ind denotes the category of ind-objects ( so the free completion under filtered colimits) and the exponential are for categories of functors.

This is proved By Lurie in Higher topos theory (proposition 5.3.5.15) when $C$ is an infinity category and $I$ is a finite poset. He gives an example to show that this cannot be generalized to the case of $I$ a finite simplicial sets - but finite category is much more restrictive and his example doesn't rule this out at all.

I'm mostly interested in the case where $C$ is a 1-category and already has finite colimits, but I'm curious of any interesting things that can be said more generally.

$\operatorname{Ind}(C^I) = \operatorname{Ind}(C)^I$?

$\DeclareMathOperator\Ind{Ind}$Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $\Ind(C^I) \to \Ind(C)^I$ is an equivalence of categories? where $\Ind$ denotes the category of ind-objects (so the free completion under filtered colimits) and the exponentials are for categories of functors.

This is proved by Lurie in Higher topos theory (proposition 5.3.5.15) when $C$ is an infinity category and $I$ is a finite poset. He gives an example to show that this cannot be generalized to the case of $I$ a finite simplicial set — but finite category is much more restrictive and his example doesn't rule this out at all.

I'm mostly interested in the case where $C$ is a 1-category and already has finite colimits, but I'm curious of any interesting things that can be said more generally.

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Simon Henry
Simon Henry
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$Ind(C^I) = Ind(C)^I$?

Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $Ind(C^I) \to Ind(C)^I$ is an equivalence of categories? where Ind denotes the category of ind-objects ( so the free completion under filtered colimits) and the exponential are for categories of functors.

This is proved By Lurie in Higher topos theory (proposition 5.3.5.15) when $C$ is an infinity category and $I$ is a finite poset. He gives an example to show that this cannot be generalized to the case of $I$ a finite simplicial sets - but finite category is much more restrictive and his example doesn't rule this out at all.

I'm mostly interested in the case where $C$ is a 1-category and already has finite colimits, but I'm curious of any interesting things that can be said more generally.