Definition of adjoint functor similar to the definition of homotopy equivalence?

Question

I am new at category theory and I haven't get the definition of adjoint functor. I have seen the another definition of natural transformation much similar to the definition of homotopy.

Other question in here gives the correlation between the basic concepts between topology and category, and it says that the adjoint functor is related to the homotopy equivalence. Thus I guess there is a definition of adjoint functor similar to the definition of homotopy equivalence, but I didn't get it.

Counit-unit adjunction seems to give most close definition of adjoint functor I wanted, but I fail to find the direct relation between the counit-unit adjunction and the homotopy equivalence.

So my question is: there is a definition of adjoint functor resembles to the definition of homotopy equivalence, as like the question in MathOverflow I mentioned as a link? I would appreciate any help.

Answer 1

The concept of adjunction is essentially a feature of 2-categories. Given a 2-category $\mathsf{C}$, two 1-morphisms $u : A \to B$ and $f : B \to A$ in $\mathsf{C}$ are said to be adjoint (denoted $u \dashv f$) when there exist 2-morphisms, the unit $\eta : \operatorname{id}_B \Rightarrow uf$ and the counit $\epsilon : fu \Rightarrow \operatorname{id}_A$ satisfying the usual triangle relations (that I won't write down here). Two adjoint functors are in some sense "dual" to each other. Both adjunction of categories are homotopy equivalences are special cases of this general notion of adjunction:

• Let $\mathsf{Cat}$ be the 2-category whose objects are categories, 1-morphisms are functors, and 2-morphisms are natural transformations. Then an adjunction as defined above is exactly the usual notion of adjunction between two functors.
• Let $\mathsf{Top}$ be the 2-category^* whose objects are topological spaces, 1-morphisms are continuous maps, and 2-morphisms are homotopies. More precisely, given two continuous maps $f,g : X \to Y$, a 2-morphism $\alpha : f \Rightarrow g$ is a continuous map $\alpha : X \times [0,1] \to Y$ such that $\alpha(-,0) = f$ and $\alpha(-,1) = g$. Then every homotopy equivalence yields an adjunction in $\mathsf{Top}$.

The analogy is, however, not perfect: in $\mathsf{Top}$, every adjunction is in fact something stronger, an adjoint equivalence: the unit and the counit are invertible (like every 2-cell in $\mathsf{Top}$, in fact), which is not the case in $\mathsf{Cat}$.

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^* This is not a strict 2-category, as vertical 2-cell composition is not strictly associative, but this is not really important here for the intuition.