There are many examples of exact functors, and also there are standard examples for contravariant/covariant left exact (e.g. hom functor) and covariant right exact (e.g. tensor product). Is there any example of contravariant right exact functor which is in general not left exact, especially the one with explicit description (i.e. not just like "adjoint to some functor in some category") or the one which is found to be useful? Also, I'd like to find some general abstract explanation on why there is no easy example for contravariant right exact functor while the other three types have typical examples.
[Math] Contravariant right exact functor
ct.category-theoryhomological-algebra
Related Solutions
The adjoint functor theorem as stated here and the special adjoint functor theorem (which can also both be found in Mac Lane) are both very handy for showing the existence of adjoint functors.
First here is the statement of the special adjoint functor theorem:
Theorem Let $G\colon D\to C$ be a functor and suppose that the following conditions are satisfied:
(i) $D$ and $C$ have small hom-sets
(ii) $D$ has small limits
(iii) $D$ is well-powered i.e., every object has a set of subobjects (where by a subobject we mean an equivalence class of monics)
(iv) $D$ has a small cogenerating set $S$
(v) $G$ preserves limits
Then $G$ has a left adjoint.
Example I think this is a pretty standard example. Consider the inclusion CHaus$\to$Top of the category of compact Hausdorff spaces into the category of all topological spaces. Both of these categories have small hom-sets, it follows from Tychonoff's Theorem that CHaus has all small products and it is not so hard to check it has equalizers so it has all small limits and the the inclusion preserves these. CHaus is well-powered since monics are just injective continuous maps and there are only a small collection of topologies making any subspace compact and Hausdorff. Finally, one can check that $[0,1]$ is a cogenerator for CHaus. So $G$ has a left adjoint $F$ and we have just proved that the Stone-Čech compactification exists.
If you have a candidate for an adjoint (say the pair $(F,G)$) and you want to check directly it is often easiest to try and cook up a unit and/or a counit and verify that there is an adjunction that way - either by using them to give an explicit bijection of hom-sets or by checking that the composites $$G \stackrel{\eta G}{\to} GFG \stackrel{G \epsilon}{\to} G$$ and $$F \stackrel{F \eta}{\to} FGF \stackrel{\epsilon F}{\to} F$$ are identities of $G$ and $F$ respectively.
I thought (although I am at the risk of this getting excessively long) that I would add another approach. One can often use existing formalism to produce adjoints (although this is secretly using one of the adjoint functor theorems in most cases so in some sense is only psychologically different). For instance as in Reid Barton's nice answer if one can interpret the situation in terms of categories of presheaves or sheaves it is immediate that certain pairs of adjoints exist. Andrew's great answer gives another large class of examples where the content of the special adjoint functor theorem is working behind the scenes to make verifying the existence of adjoints very easy. Another class of examples is given by torsion theories where one can produce adjoints to the inclusions of certain subcategories of abelian (more generally pre-triangulated) categories by checking that certain orthogonality/decomposition properties hold.
I can't help remarking that one instance where it is very easy to produce adjoints is in the setting of compactly generated (and well generated) triangulated categories. In the land of compactly generated triangulated categories one can wave the magic wand of Brown representability and (provided the target has small hom-sets) the only obstruction for a triangulated functor to have a right/left adjoint is preserving coproducts/products (and the adjoint is automatically triangulated).
To speak of right (or left) adjoints for a contravariant functor $F: C\to D$, one needs to decide whether to view it as a functor from $C^{op}$ to $D$ or as a functor from $C$ to $D^{op}$. What the one viewpoint calls a left adjoint, the other will call a right adjoint. One therefore often speaks instead of two contravariant functors being "adjoint on the right" (or on the left). In this language, $Hom(-,A)$ is adjoint on the right to itself, which means that morphisms from $X$ to $Hom(Y,A)$ are in natural bijective correspondence with morphisms from $Y$ to $Hom(X,A)$; here "natural" means (as usual for adjointness) with respect to both $X$ and $Y$.
Best Answer
There is a natural functor with such property in the theory of coalgebras and co/contramodules over them. Given a (coassociative, counital) coalgebra $C$ over a field $k$, a left comodule $M$ over $C$ is a $k$-vector space with a structure (coaction) map $M\to C\otimes_k M$. A left contramodule $P$ over $C$ is a $k$-vector space with the structure (contraaction) map $Hom_k(C,P)\to P$. (The appropriate co/contraassociativity and counit axioms have to be satisfied in both cases.)
Given a left $C$-comodule $M$ and a left $C$-contramodule $P$, the $k$-vector space of cohomomorphims $Cohom_C(M,P)$ is defined as the quotient space of the vector space $Hom_k(M,P)$ by the image of the difference of two maps $Hom_k(C\otimes_k M,P)\rightrightarrows Hom_k(M,P)$, one of which is induced by the $C$-coaction in $M$ and the other one by the $C$-contraaction in $P$. The $Cohom$ is a right exact functor in both of its arguments, contravariant in the first (comodule) argument and covariant in the second one.