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If $F\colon\mathcal{A}\to\mathcal{B}$ is left adjoint to $U\colon\mathcal{B}\to\mathcal{A}$, then $U$ preserves limits and $F$ preserves colimits.

Can we say something more if $F$ is left adjoint to $U$ and they form an adjoint equivalence between $\mathcal{A}$ and $\mathcal{B}$?

My question arises from the following statement on Wikipedia:

The functor $H\colon\mathcal{I}\to\mathcal{C}$ has limit (or colimit) $\mathcal{I}$ if and only if the functor $FH\colon\mathcal{I}\to\mathcal{D}$ has limit (or colimit) $F\mathcal{I}$.

where $F\colon\mathcal{C}\to\mathcal{D}$ is an equivalence. Is $F$ preserving both limits and colimits? (in the Wikipedia case, $F$ is not part of an adjoint equivalence, but, according to this question, every equivalence can be upgraded to an adjoint equivalence).

Arnaud D.
  • 20,884

2 Answers2

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An adjoint equivalence between categories $\mathcal{A}$ and $\mathcal{B}$ is not a pair $(F,U)$ of functors between them, but a quadruplet $(F,U,\eta,\varepsilon)$, where $\eta$ and $\varepsilon$ are corresponding natural transformations, such that $(F,U,\eta,\varepsilon)$ is an adjunction. Note, that every pair $(F,U)$, where $F$ and $U$ are mutually inverse equivalences of categories, may be completed to such a quadruplet. Therefore, every equivalence of categories is both a left-adjoint and a right-adjoint functor.

Of course, equivalences of categories preserve much more constructions than left-adjoints or right-adjoints. Actually, they preserve "all" categorical information (the things not preserved by equivalences are half-jokingly called "evil"). In particular, equivalences preserve and reflect all limits and colimits (they preserve limits because they are right-adjoints, preserve colimits because they are left-adjoints, reflect limits and colimits because they are fully faithful).

Oskar
  • 4,363
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When $F$ and $U$ form an adjoint equivalence, the unit $\eta:I_{\mathcal{A}}\to UF$ and counit $\epsilon : FU\to I_{\mathcal{B}}$ are isomorphisms; then their inverses $\epsilon^{-1}:I_{\mathcal{B}}\to FU$ and $\eta^{-1}:UF\to I_{\mathcal{A}}$ are natural transformation, and they satisfy the identities $$F(\eta^{-1})\circ\epsilon^{-1}_F=(\epsilon_F\circ F(\eta))^{-1}=1_F$$ and $$\eta^{-1}_U\circ U(\epsilon^{-1})=(U(\epsilon)\circ \eta_U)^{-1}=1_U.$$ Thus $U$ is also left adjoint to $F$, so that $U$ preserve colimits and $F$ preserve limits.

Arnaud D.
  • 20,884