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There is a way in which we can think about a natural transformation $\eta: F \rightarrow G$ as a homotopy between functors $F,G:\mathcal{C}\rightarrow \mathcal{D}$. Now, an adjunction $F \dashv G$ gives rise to "homotopies" $\epsilon:FG \rightarrow 1_{\mathcal{C}}$ and $\eta:1_{\mathcal{D}} \rightarrow GF$, namely the counit and unit. If again we borrow the language of topology, one could say that the existence of an adjunction implies that the two categories $\mathcal{C}$ and $\mathcal{D}$ have the same "homotopy type".

Is this a useful picture to have in mind when dealing with adjunctions? Is it well-known?

Thanks for your time.

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  • $\begingroup$ @ThomasAndrews I meant natural transformation! $\endgroup$ – Alex Provost Apr 23 '13 at 19:42
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    $\begingroup$ "Is it well-known?" oh yes, algebraic topologists use this all the time! $\endgroup$ – Martin Brandenburg Apr 24 '13 at 7:41
  • $\begingroup$ @MartinBrandenburg Thanks, this is good to know :-) $\endgroup$ – Alex Provost Apr 24 '13 at 14:54
  • $\begingroup$ A similar question. $\endgroup$ – Najib Idrissi Oct 5 '15 at 12:20
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Yes. An adjunction between two categories gives rise to a homotopy equivalence between their nerves.

Also observe that if $I = \{ 0 \to 1 \}$ denotes the interval category, then a natural transformation between two functors is precisely a functor $C \times I \to D$. So natural transformations are like homotopies between functions in this sense, except that they are "directed" in a way that homotopies aren't. This idea gives rise to directed homotopy theory.

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  • $\begingroup$ Interesting! I didn't think about the "directed" part of the homotopy. A classical homotopy is symmetric in terms of the two maps involved (in the sense that $f \simeq g \iff g \simeq f$), but a natural transformation between two functors does not necessarily give rise to an "opposite" transformation. $\endgroup$ – Alex Provost Apr 23 '13 at 19:51
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    $\begingroup$ Yes, that's the important difference. If $C, D$ are groupoids then any adjunction is automatically a natural equivalence (we have removed any directedness, and in fact groupoids are much more closely related to homotopy theory since they model homotopy $1$-types). $\endgroup$ – Qiaochu Yuan Apr 23 '13 at 19:54
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    $\begingroup$ I would say that in general homotopy is not an equivalence relation, except for nice objects. projecteuclid.org/DPubS/Repository/1.0/… $\endgroup$ – Baby Dragon Apr 23 '13 at 20:42
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    $\begingroup$ Homotopy, defined in the classical sense for topological spaces using the standard interval, is an equivalence relation, and it is clear that is what the OP is thinking about. (Also note that in the model structure on $\textbf{Cat}$, two functors are homotopic precisely if they are isomorphic, not merely linked by a natural transformation.) $\endgroup$ – Zhen Lin Apr 23 '13 at 21:48

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