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• CommentRowNumber1.
• CommentAuthorMirco Richter
• CommentTimeApr 16th 2012
• (edited Apr 16th 2012)

Suppose you have a category $C$ and a category $D$ such that $C$ is a subcategory of $D$ and $D$ is a subcategory of $\mathrm{Set}$. Moreover the simplicial skeleton (endo)functor ${\mathrm{Sk}}^{n}$
exists in the category of simplicial $C$ objects $\mathrm{sC}$ but not in the category of simplicial $D$ objects $\mathrm{sD}$.

Moreover suppose $X\in \mathrm{sC}$ is ’$n$-skeleton’ (means ${\mathrm{Sk}}^{n}\left(X\right)=X$). Then there is the adjunction

${\mathrm{Hom}}_{\mathrm{sC}}\left(X,Y\right)\simeq {\mathrm{Hom}}_{\mathrm{sC}}\left(X,{\mathrm{Cosk}}^{n}Y\right)$ for any $Y\in \mathrm{sC}$.

Now suppose that the endofunctor ${\mathrm{Cosk}}^{n}$ still exsists in $\mathrm{sD}$ and that now $Y\in \mathrm{sD}$ but $Y\notin \mathrm{sC}$.

Does the adjunction ${\mathrm{Hom}}_{\mathrm{sD}}\left(X,Y\right)\simeq {\mathrm{Hom}}_{\mathrm{sD}}\left(X,{\mathrm{Cosk}}^{n}Y\right)$ still holds?

Something like this occurs for example when $C$ is the category of vector spaces and $D$ is the category of Banach manifolds and $X$ is a simplicial vector space with ${\mathrm{Sk}}^{n}X=X$ and $Y$ is a ’nonlinear’ simplicial manifold.

• CommentRowNumber2.
• CommentAuthorMirco Richter
• CommentTimeApr 17th 2012

nothing?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 17th 2012
• (edited Apr 17th 2012)

Moreover suppose $X\in \mathrm{sC}$ is ’$n$-skeleton’

One says $X$ is $n$-skeletal.

I think you are after a relative adjoint functor (seel local definition of adjoints).

To answer your question, you will need some assumption on the inclusion $C\to D$ preserving the colimits that $\mathrm{sk}$ is built from. If they are preserved (and I take it you assume that the dual limits that $\mathrm{cosk}$ is built from exist in $D$), then the desired relation will hold.

• CommentRowNumber4.
• CommentAuthorMirco Richter
• CommentTimeApr 19th 2012

No the relative adjoint functor definition doesn’t apply here. But ok, In that case I think the answer is just that you can’t generally say that the natural isomorphism exists…

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 19th 2012

No the relative adjoint functor definition doesn’t apply here.

Here is how:

Apply the following dictionary to translate from the notation at relative adjoint functor to your notation

• $C$ becomes $\mathrm{sD}$

• $D$ becomes $\mathrm{sD}$

• $B$ becomes $\mathrm{sC}$

• $J:B\to D$ becomes $\mathrm{sC}↪\mathrm{sD}$.

• $R:C\to D$ becomes ${\mathrm{cosk}}_{n}:\mathrm{sD}\to \mathrm{sD}$

• $L:B\to C$ becomes ${\mathrm{sk}}_{n}:\mathrm{sC}\to \mathrm{sD}$.

With that the natural iso

${\mathrm{Hom}}_{C}\left(L\left(-\right),-\right)\simeq {\mathrm{Hom}}_{D}\left(J\left(-\right),R\left(-\right)\right)$

becomes the natural iso that you are after

${\mathrm{Hom}}_{\mathrm{sD}}\left({\mathrm{sk}}_{n}\left(X\right),\left(Y\right)\right)\simeq {\mathrm{Hom}}_{\mathrm{sD}}\left(X,{\mathrm{cosk}}_{n}Y\right)$

for $X\in \mathrm{sC}$, $Y\in \mathrm{sD}$

• CommentRowNumber6.
• CommentAuthorMirco Richter
• CommentTimeApr 19th 2012
• (edited Apr 19th 2012)

I know, but what I mean is that this is not a solution to the problem. Just a formal clean way to write it down. We only now that the natural iso exists in the subcategory $\mathrm{sC}$, but if I understand it right this doesn’t implay

${\mathrm{Hom}}_{\mathrm{sD}}\left({\mathrm{sk}}_{n}\left(X\right),Y\right)\simeq {\mathrm{Hom}}_{\mathrm{sD}}\left(X,{\mathrm{cosk}}_{n}Y\right)$ for

$X\in \mathrm{sC}$ and $Y\in \mathrm{sD}$. Right?

So I still ave to show that the iso exist here.

• CommentRowNumber7.
• CommentAuthorMirco Richter
• CommentTimeApr 19th 2012

If someone knows more please let me know. (I know most of the time I’m here for just asking questions, but you are the leading experts :-) and those things are quite cumbersome to me). The question is two fold:

Assuming I know that in $\mathrm{sC}$ the isomorphism ${\mathrm{Hom}}_{\mathrm{sC}}\left({\mathrm{sk}}_{n}\left(X\right),Y\right)\simeq {\mathrm{Hom}}_{\mathrm{sC}}\left(X,{\mathrm{cosk}}_{\mathrm{nY}}\right)$ exists.

1.) Do I still have to prove that there is a natural iso giving ${\mathrm{Hom}}_{\mathrm{sD}}\left({\mathrm{sk}}_{n}\left(X\right),Y\right)\simeq {\mathrm{Hom}}_{\mathrm{sD}}\left(X,{\mathrm{cosk}}_{n}Y\right)$ for $X\in \mathrm{sC}$ and $Y\in \mathrm{sD}$, or is this automatic?

2.) If I still have to prove it, can I take somehow advantage of the fact that the isomorphism exist in the subcat $\mathrm{sC}$?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 19th 2012

As I said, you need that the inclusion preserves the colimits.

• CommentRowNumber9.
• CommentAuthorMirco Richter
• CommentTimeApr 20th 2012

Ok. Just a little fine tuning:

1.) Must $J$ preserve all colimits or only those used in ${\mathrm{sk}}_{n}$?

2.) Must $D$ have all limits or only those used in ${\mathrm{cosk}}_{n}$?