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Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. We have an associated pair of functors $\tilde{F}:{\mathrm{Simp}}C\to {\mathrm{Simp}}D$ and $\tilde{G}:{\mathrm{Simp}}D\to {\mathrm{Simp}}C$ between these simplicial categories. Is $\tilde{F}$ and $\tilde{G}$ still adjoint functors, at least up to homotopy?

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up vote 6 down vote accepted

Yes.

First something more general:

Let $A$ be a small category and let $F: C \leftrightarrow D : G$ be an adjoint pair of functors. Then there is an associated adjoint pair of functors $F^A : C^A \leftrightarrow D^A:G^A$ on the respective categories $C^A$ and $D^A$ of functors $A \to C$ and $A\to D$

To prove this, check:

  1. Every functor $H: C \to D$ gives a functor $H^A: C^A \to D^A$ (simply by postcomposition of a functor $A \to C$ with $H:C \to D$).
  2. Every natural transformation $\alpha: H \Rightarrow H'$ between functors $H,H': C \to D$ gives a natural transformation $\alpha^A : H^A \Rightarrow H'{}^A$.
  3. Unit and counit of the adjunction $F: C \leftrightarrow D: G$ extend to unit and counit of an adjunction $F^A:C^A \leftrightarrow D^A:G^A$: verify the triangle identities (called counit-unit equations on Wikipedia).

Now observe that a simplicial object in $C$ is the same as a contravariant functor $\Delta \to C$, where $\Delta$ is the simplex category consisting of the finite ordinal numbers and order-preserving maps, so $\operatorname{Simp}{C} = C^{\Delta^{\rm op}}$.

Apply the above with $A = \Delta^{\rm op}$.

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Theo, nice answer! I found it interesting in its generality (I don't know anything about simplicial stuff). One question, though. Modulo size issues, you have "applied the internal hom functor $\hom_{\textbf{CAT}}(A,-):\textbf{CAT} \to \textbf{CAT}$ to the adjunction given by $(F,G)$", and you gave a sketch of the proof that it is still an adjunction. Is this still true of any functor $X:\textbf{CAT}\to \textbf{CAT}$? Thanks. –  Bruno Stonek Jul 31 '12 at 14:03
    
@Bruno: Thanks, however, I'm not sure I understand what exactly you're asking: an endo-functor of the category $\mathbf{CAT}$ doesn't necessarily give you anything on the natural transformations (that Hom does give you that is used here in point 3). So I'd expect a similar statement as soon as $X$ is a $2$-functor with the appropriate strictness-properties but I don't feel comfortable enough with higher stuff to say anything definitive. An example of interest might help me see what you want, but I'm not making any guarantees. –  t.b. Aug 1 '12 at 19:36
    
I think you perfectly understood what I was asking. I guess I'll look up on 2-functors. Thank you! –  Bruno Stonek Aug 1 '12 at 19:43
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