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# CocompleteAnycocomplete category (s.t....)withadensesmallfullsubcategory is complete?

Hello,

I with to consider the following statement:

If $C$ is a cocomplete category having a dense small full subcategory $D$, then $C$ is complete.

(a full subcategory $D$ is dense in $C$ if every element of $C$ is canonical colimit of elements of $D$...)

I think I know how to prove it (I give proof below), and I want someone to reassure me that this statement is true exactly as stated, as it seems a little bit surprising.

Proof sketch:

Consider the functor $Y : C \to psh(D)$, where $psh(D)$ denotes the category of presheaves on $D$ (Yoneda functor, i.e. $Y(X)(A) = Hom (A, X)$). $D$ being dense in $C$ is equivalent to this functor being fully faithful. Futhermore, we have a functor $F: psh(D) \to C$, namely, the one which extends the inclusion $D \to C$ by cocontinuity (as presheaf categories have the property of being free cocompletions: http://ncatlab.org/nlab/show/free+cocompletion). Then one can see that $Y$ is right adjoint to $F$. So this renders $C$ as reflective subcategory of $psh(D)$. psh(D)$( http://ncatlab.org/nlab/show/reflective+subcategory). Now,$psh(D)$is complete, and so every reflective subcategory of it. hence,$C\$ is complete.

Thank you, Sasha

1

# Cocomplete category (s.t. ...) is complete

Hello,

I with to consider the following statement:

If $C$ is a cocomplete category having a dense small full subcategory $D$, then $C$ is complete.

(a full subcategory $D$ is dense in $C$ if every element of $C$ is canonical colimit of elements of $D$...)

I think I know how to prove it (I give proof below), and I want someone to reassure me that this statement is true exactly as stated, as it seems a little bit surprising.

Proof sketch:

Consider the functor $Y : C \to psh(D)$, where $psh(D)$ denotes the category of presheaves on $D$ (Yoneda functor, i.e. $Y(X)(A) = Hom (A, X)$). $D$ being dense in $C$ is equivalent to this functor being fully faithful. Futhermore, we have a functor $F: psh(D) \to C$, namely, the one which extends the inclusion $D \to C$ by cocontinuity. Then one can see that $Y$ is right adjoint to $F$. So this renders $C$ as reflective subcategory of $psh(D)$. Now, $psh(D)$ is complete, and so every reflective subcategory of it. hence, $C$ is complete.

Thank you, Sasha
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