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I have a question about the definition of the (left) adjoint of a functor.I am trying to understand the philosophy and reason of the definition of adjoint functors.

If I understand correctly the situation is as follows: Suppose one has the category $\mathcal{C}$ with an object $A$ and a category $\mathcal{D}$ with an object $B$ and functors $F: \mathcal{C} \to \mathcal{D}$ and a functor $G: \mathcal{D} \to \mathcal{C}$.

Now one can wonder how to make kind of cross morphism from $A$ to $B$. The two natural ways would be to push $A$ to $\mathcal{D}$ using $F$ and take a morphism $\phi$ there. But there is another equally natural way to this. Push $B$ to $\mathcal{C}$ by $G$ and take a morphism $\psi$ between $A$ and $GB$.

The natural equivalence then translates approach one in to approach two and back by making a correspondence between $\phi$ and $\psi$.

To make it a bit sensible one demands that the functors $F$ and $G$ are not arbitrary, but somehow linked together in the sense that linking $A_1 \to A$ with $B \to B_1$ is done on essentially( up to correspendance by a unit) the same way by pushing $A$ to $\mathcal{D}$ or pushing $B$ to $\mathcal{C}$.

Is this indeed the philosophy behind adjoint functors?

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Thanks @usero2139 for correcting my spelling mistakes. A bit of my native langue Dutch that was seeping trough. – MrOperator Jan 3 '12 at 14:22
You could view it that way if you like. But the usual motivation behind adjunctions is that they capture the idea of "most universal solution to a problem". See here. – Zhen Lin Jan 3 '12 at 14:24
up vote 10 down vote accepted

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

Usually when 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 condition 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 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 of the category $\mathbf A$, and the opposite. This enables us to transport problems in the most convenient setting.

I hope this answer was useful.

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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.

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