Can we think of an adjunction as a homotopy equivalence of categories?

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?

• @ThomasAndrews I meant natural transformation! Apr 23, 2013 at 19:42
• "Is it well-known?" oh yes, algebraic topologists use this all the time! Apr 24, 2013 at 7:41
• @MartinBrandenburg Thanks, this is good to know :-) Apr 24, 2013 at 14:54
• A similar question. Oct 5, 2015 at 12:20

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.

• 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. Apr 23, 2013 at 19:51
• 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). Apr 23, 2013 at 19:54
• I would say that in general homotopy is not an equivalence relation, except for nice objects. projecteuclid.org/DPubS/Repository/1.0/… Apr 23, 2013 at 20:42
• 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.) Apr 23, 2013 at 21:48