Philosophy or meaning of adjoint functors

Solution 1:

As the link of Zhen Lin says adjoint functors deal with universals.

Usually I think to adjointness situations in two different ways: as conditions of global existence of universals (related to the comment of Zhen Lin above) and as a conditions of global representability. (By Yoneda lemma these things are essentially the same but they are different, from a psychological point of view).

Let's see the first: the global existence of universals.

The following fact holds: a functor $\mathcal G \colon \mathbf A \to \mathbf X$ has a left adjoint if and only if for each $x \in \mathbf X$ exists a morphisms $\eta_x \colon x \to \mathcal G(\mathcal F(x))$, where $\mathcal F(x) \in \mathbf A$, which is universal from $x$ to $\mathcal G$ [i.e. for every other object $a \in \mathbf A$ and morphism $\tau \colon x \to \mathcal G(a)$ exists a unique $h \colon \mathcal F(x) \to a$ such that $\mathcal \tau=G(h)\circ \eta_x$].

In this case it can be show that the induced function $\mathcal F$ from the set of objects of $\mathbf X$ to the the set of objects of $\mathbf A$ can be extended to a functor $\mathcal F \colon \mathbf X \to \mathbf A$ (unique up to natural isomorphism). This condition of adjointness, the existence of one universal morphism from every object in $\mathbf A$, justifies the phrase global existance of universals.

There's also a dual result that says that a functor $\mathcal F \colon \mathbf X \to \mathbf A$ has a right adjoint $\mathcal G \colon \mathbf A \to \mathbf X$ if and only if exists a family of morphisms $\epsilon_a \colon \mathcal F(\mathcal G(a)) \to a$ each one being universal from $\mathcal F$ to $a \in \mathbf A$, i.e. for each other morphisms $\sigma \colon \mathcal F(x) \to a$ exists a unique $k \colon x \to \mathcal G(a)$ such that $\sigma = \epsilon_a \circ \mathcal F(k)$.

These facts can be rephrased in term of representable functors. So the existance of the family of universal morphisms $\langle \eta_x \colon x \to \mathcal G(\mathcal F(x))\rangle_{x \in \mathbf X}$ is equivalent via Yoneda's lemma to the representability, meaning that the functor $\mathcal G$ has a left adjoint if each functor $\mathbf X(x,\mathcal G(-)) \colon \mathbf A \to \mathbf {Set}$ is representable [that's because each universal morphism $\eta_x \colon x \to \mathcal G(\mathcal F(x))$ gives a natural isomorphism $\mathbf X(x,\mathcal G(-)) \cong \mathbf A(\mathcal F(x),-)$]. Similarly the existance of the family of universal morphisms $\langle \epsilon_a \colon \mathcal F(\mathcal G(a)) \to a\rangle_{a \in \mathbf A}$ implies that each of the functors $\mathbf A(\mathcal F(-),a)$ is representable [because the universal morphism $\epsilon_a \colon \mathcal F(\mathcal G(a)) \to a$ give a natural isomorphism $\mathbf A(\mathcal F(-),a) \cong \mathbf X(-,\mathcal G(a))$].

Representability says that each fact about morphisms of type $x \to \mathcal G(a)$ is equivalent to a fact about morphisms of type $\mathcal F(x) \to a$. Simplifying this means that we can traduce facts/problems in the category $\mathbf X$ in other facts/problems in the category $\mathbf A$, and the opposite. This enables us to transport problems in the most convenient setting.

I hope this answer was useful.

Solution 2:

One should also mention the utility of adjoint functors for explicit computation, which happens because left adjoints preserve colimits and right adjoints preserve limits.

Consider for example the forgetful functor $Ob$ from groupoids to sets. This has a right adjoint, which to a set $X$ assigns what is called sometimes the indiscrete or coarse groupoid, say $Ind(X)$, on $X$, which has exactly one morphism between any two objects.

Now suppose we want to compute $L=$colim$G_i$ of some diagram of groupoids. Because of the above, we know that $Ob(L)=$colim$Ob(G_i)$. That is a start.