Fundamental Adjunctions

In this lightheaded post (written long time ago) I want to share with you some fundamental adjunctions that are the "source" of various other adjunctions that pop up all over in mathematics (well, at least all over algebraic topology).

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A baby case

Let $A,B,C$ be sets, and denote by $Map(A,B)$ the set of set-theoretic maps $A \to B$. Then
$! Map(A \times B, C) \cong Map(A, Map(B,C)).$ This is sometimes called Currying or Schönfinkeln, and it really boils down to
$! [ (a,b) \mapsto f(a,b) ]\quad \mapsto\quad [a \mapsto [b \mapsto f(a,b)] ].$

Definition

An adjunction is a pair of functors $F : C \to D$, $G : D \to C$ between categories $C,D$ together with a natural transformation $\alpha$ of functors $C \times C \to Sets$
$! \alpha : Hom_D(F-,-) \Rightarrow Hom_C(-,G-).$ In the baby case above, $C=Sets$ and $F: A \mapsto A\times B$ is adjoint to $G : C \mapsto Map(B,C)$.
We call $F$ the left adjoint to $G$ and $G$ the right adjoint to $F$. The notation $F$ ⊣ $G$ can be quite useful.

You should care about adjunctions, because their existence implies nice properties: $F$ is always cocontinuous (preserves colimits) and $G$ is always continuous (preserves limits). This comes from the continuity properties of $Hom(-,-)$. There is also a theorem (Freyd's AFT) that describes precisely which extra condition a functor has to satisfy, besides continuity, to be a right adjoint.

Another example comes from vector spaces:
$! Hom(V \otimes W, X) \cong Hom(V,Hom(W,X))$ where we give $Hom(W,X)$ the vector space structure via $X$.
In fact, this is basically the definition of the tensor product, if you watch closely.

In homotopy theory, one often uses the adjunction
$! [\Sigma -,-] \cong [-,\Omega -]$ between suspension $\Sigma$ and loop space $\Omega$ (where the brackets $[,]$ denote the morphisms in the homotopy category). From the definitions, $\Sigma X$ is a quotient of $X \times [0,1]$ and $\Omega X$ is a subspace of $X^{[0,1]}$, and the source of this fundamental adjunction is the baby case above.

Forgetting Adjunctions

To a lot of "forgetful" functors, like from rings with unit to rings not-necessarily-with-unit, from abelian groups to arbitrary groups, from abelian groups to abelian monoids, from compact Hausdorff spaces to arbitrary topological spaces, one can construct left adjoints which give some sort of "free" objects. For example, to any topological space one can assign the Stone-Cech compactification, and to every abelian monoid one can assign the Grothendieck group.

Limits and Colimits

For categories $C,J$ we can form the functor category $C^J$ of functors $J \to C$ and consider the diagonal functor $\Delta : C \to C^J$ which maps every object to the constant functor. If it has a right adjoint, then we call it Limit ($lim : C^J \to C$) and write
$! Hom_{C^J}(\Delta -, -) \cong Hom_{C}(-, lim - ).$
If it has a left adjoint, then we call it Colimit ($colim : C^J \to C$) and write
$! Hom_{C}(colim -, -) \cong Hom_{C^J}(-, \Delta - ).$
Such a (co)limit functor exists iff all (co)limits exist in the ordinary sense.

There would be much more to write (e.g. about nerves and realizations) but I'll stop here to prevent this from rotting in the draft section.

Except, I couldn't resist [UPDATE 2014-06-12]:

The nerve of a category

Given a category $C$ we can look at all chains of arrows of length $n$, i.e. $f_1,\dots,f_n$ such that one has a composition $f_n \circ \cdots \circ f_1$. Such a chain of length $n$ can be mapped to a chain of length $n-1$ by merging two adjacent arrows $f_i,{f_i+1}$ into $f_{i+1}\circ f_i$ and it can be mapped to a chain of length $n+1$ by inserting identities. All this structure together forms a simplicial set $NC$ called the nerve, whose $n$-simplices are precisely the chains of arrows of length $n$. This construction if functorial, and it admits a left adjoint $h$ that associates to a simplicial set its "homotopy category":
$! Hom_{Cat}(h -, -) \cong Hom_{sSet}(-, N -).$

A better (well...) explanation of the suspension-loop space adjunction

Write $Ho$ for the homotopy category of a category $C$ with weak equivalences, then we have a derived version of the lim and colim adjunctions:
$! Ho_{C^J}(\Delta -, -) \cong Ho_{C}(-, holim - ),$
$! Ho_{C}(hocolim -, -) \cong Ho_{C^J}(-, \Delta - ).$
Specializing this to the diagrams $J$ being $\cdot \rightarrow \cdot \leftarrow \cdot$ (for holim) and $\cdot \leftarrow \cdot \rightarrow \cdot$ (for hocolim) we are talking about homotopy pullbacks and homotopy pushouts. Now plugging in a terminal object $\ast$ for the outer things, and an arbitrary object $X$ in the middle, we get diagrams in $C$ like $\ast \rightarrow X \leftarrow \ast$, whose holim is called $\Omega X$ and $\ast \leftarrow X \rightarrow \ast$ whose hocolim is called $\Sigma X$.

At this point you should probably try to see that the ordinary suspension and loop space fulfill these universal properties in the category of topological spaces.

Anyway, writing down explicitly the holim and hocolim adjunctions for these particular diagrams, you see the adjunction immediately. I guess this is called "synthetic homotopy theory" and it's beautiful, isn't it?