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Let $C,D$ be two presentable categories, $F,G : C \to D$ cocontinuous functors and $\eta :F \to G$ be a morphism of functors. Is it always true that the full subcategory

$\{x \in C : F(x) \xrightarrow{\eta(x)} G(x) \text{ is an isomorphism}\}$

of $C$ is presentable? In fact I only want to know if this category is complete (which happens to be the case in many examples; where of course the inclusion to $C$ doesn't have preserve limits). If necessary, you may assume that $F,G$ preserve finite limits.

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This seems to be the strict 2-limit known as an "inverter". Exercise 2.m in LPAC says that inverters are accessible, so that's halfway to being presentable! –  Zhen Lin Nov 29 '12 at 9:08
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Ok but cocomplete+accessible=presentable. So the answer seems to be Yes. Anyway it would be great if someone could explain this in an answer more directly (without the whole theory of chapter 2 in LPAC). –  Martin Brandenburg Nov 29 '12 at 9:25
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Out of curiosity, why are you calling them "presentable"? I thought the standard term, even if it's not a great term, is "locally presentable". –  Todd Trimble Nov 29 '12 at 14:57
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Yes, the "locally" refers to the fact that it's the objects that are being presented. (Actually, there are perfectly reasonable senses in which we can speak of the categories themselves ("globally") presented, as in a topos of sheaves presented by a site.) –  Todd Trimble Nov 30 '12 at 2:34
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If your last comment was intended as a honest thankful note, you did not manage to avoid appearing slightly annoyingly sarcastic. –  Mariano Suárez-Alvarez Dec 7 '12 at 18:10

1 Answer 1

Since $\{x\in C : \eta(x) \textrm{ is an isomorphism}\}$ is cocomplete (indeed closed under colimits in $C$) it remains to show that it is accessible.

First observe that the arrow category $D^{\cdot\to\cdot}$ is again locally presentable. Let $D^{\cdot\to\cdot}_\cong$ be its full replete subcategory determined by all isomorphisms. This subcategory is also locally presentable and closed under colimits in $D^{\cdot\to\cdot}$.

Now consider the functor $H:C\longrightarrow D^{\cdot\to\cdot}$ given by $H(x) = \eta(x)$ on objects and $H(u:c\to c') = (Fu,Gu):\eta(c)\to\eta(c')$ on morphisms. Then $H$ is again a cocontinuous functor between locally presentable categories and $\{x\in C : \eta(x) \textrm{ is an isomorphism}\}$ is exactly the full preimage of $D^{\cdot\to\cdot}_\cong$ under $H$. By Remark 2.50 of LPAC it is therefore also accessible.

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