Is there a connection between free–forgetful adjunctions and tensor-hom adjunctions?

Question

In the Wiki article on adjunction https://en.wikipedia.org/wiki/Adjoint_functors, there is a motivation section that talks about how adjunctions can be viewed as "Solutions to optimization problems", or "a way of giving the most efficient solution to some problem via a method which is formulaic". The subsequent discussion, and later definition spawned from this motivation do make sense in terms of the free-forgetful adjunctions, like yes, I see how the free group construction is the "most efficient way to create a group out of a set", and hence is the left adjoint of the forgetful functor, which is the "most efficient way to create a set out of a group".

Another example of this "optimization motivation" in a situation that is not exactly a free-forgetful adjunction is the diagonal functor being adjoint to the product --- there's still some intuition there about $X \mapsto (X,X)$ being the "most natural/efficient way" of going from one object to a pair. For situations involving both left and right adjoints, the Wiki page has a very nice interpretation of this as well:

The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves. This gives the intuition behind the fact that adjoint functors occur in pairs: if F is left adjoint to G, then G is right adjoint to F.

A very nice example from A bestiary about adjunctions is the idea that quantifiers can be thought of as adjoints to the inverse image function. The inverse image functor applied to $f: X\to Y$ can be thought of as "the shadow of $Y$ on $X$ when projected through $f$", so it is reasonable that the "most efficient solution" to this problem is just to find all points of $Y$ that could have (existential quantifier) cast that shadow. Not sure yet how to phrase the right adjointness of the universal quantifier. Anyhow, many examples in the above "bestiary about adjunctions" link have some intuitive sense of being an "efficient and formulaic conversion" ...but I have no sense of this intuition at all regarding the tensor-hom adjunction.

My question is this: is there some interpretation of the tensor-hom adjunction https://en.wikipedia.org/wiki/Tensor-hom_adjunction that is in a similar line of thinking to this "optimization motivation"? I suppose implicitly my question is asking for a more formal/rigorous understanding of what the phrase "the problem $G$ poses" means in the Wiki page's sentence "$F$ is the most efficient solution to the problem posed by $G \iff F$ is the left adjoint of $G$".

EDIT 12/19/22: Indeed, trouble understanding "in which sense a functor poses a question" has already been asked on MO to the writer of the Wikipedia article, with no answer.

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EDIT 12/18/22: I came across https://mathoverflow.net/questions/6551/what-is-an-intuitive-view-of-adjoints-version-1-category-theory, with I think this answer being the one I like more than the others; in part it reads:

In essence, if two notions are related to eachother closely enough that giving a definition of one notion (in the presence of sufficient ambient structure) defines the other notion completely, they can be expressed as a (possibly $\infty$-) adjoint pair of functors. Conversely, when we see that two notions can be expressed as an adjoint pair of functors it means we can think about one by thinking about the other plus some canonical additional structure.

(this makes some sense for e.g. the Group-Set forgetful-free functor; going from Group to Set we have no canonical additional structure to add, but going back the canonical additional structure is the group structure formed by the words generated by the letters --- although I still don't really see the hom functor being "the tensor functor plus some canonical additional structure" or vice versa). I also think this comment (about the inclusion $\iota: \mathbb Z \to \mathbb R$ having right/left adjoint floor/ceil function resp.) is cute:

The right adjoint is conservative, safe, it does not want to over estimate, hence the floor [rounding down]. The left adjoint is liberal, risky, it wants to make sure it gets credit for partial work, hence the ceil [rounding up].

See also this MO thread about the prevalence of forgetful functors having left adjoints.

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Maybe one key difficulty in connecting the 2 concepts is that forgetful-free adjunctions are most intuitive through the unit-counit definition of adjunction; while the tensor-hom adjunction is most intuitive through the natural isomorphism of hom functors perspective.

EDIT 12/20/22: the previous statement is too subjective (as pointed out in the comments), and I'm starting to change my mind about it. However, another key difficulty may be that (the most common examples of) forgetful-free adjunctions are between two "tangibly" different categories (Group vs. Set for instance), so the free group generated by a set $X$ can be thought of as a "groupified version of the set $X$" in the precise sense that the Group-maps outward from the groupification $F(X)$ "are essentially" Set-maps outward from $S$ to a restricted class of sets, namely those that are "set versions" of groups. So $F(X)$ and $X$ are really the "same object" in different categories, at least in the eyes of all other groups $Y$ (and their alter egos $G(Y)$ in Set). This "outsider POV" on $F(X)$ and $X$ somehow encodes the "ambient structure" of Group and Set resp. discussed in the abovementioned MO answer.

The difficulty of the tensor-hom adjunction is perhaps then that because the categories the functors map between are similar, there is no intuition for what the "ambient structure" is. Qiaochu's answer on reframing adjunction in terms of representability seems to say that the "outsider POV" perspective is the "right" one to take, and so the fact that in the forgetful-free setting the outsider POV somehow encodes the ambient structure of each category is a "coincidence", and not really the heart of the matter.

Answer 1

Let $R, S$ be two rings and $f : R \to S$ a ring homomorphism. Pullback along $f$ induces a forgetful functor $\text{Mod}(S) \to \text{Mod}(R)$ from left $S$-modules to left $R$-modules, and this functor has a left adjoint $\text{Mod}(R) \to \text{Mod}(S)$ given by the tensor product $S \otimes_R (-)$, giving an adjunction which is simultaneously a free-forgetful and a tensor-hom adjunction:

$$\text{Hom}_S(S \otimes_R M, N) \cong \text{Hom}_R(M, \text{Hom}_S(S, N)).$$

Here $\text{Hom}_S(S, N)$ is a slightly confusing name for the forgetful functor; we are thinking of $S$ as an $(S, R)$-bimodule here. The left adjoint here is variously called change of rings, induction (in the sense of induced representations), or extension of scalars. It specializes, for example, to induced representations for groups if we take $R, S$ to be group algebras and $f$ to be a morphism induced by a group homomorphism, as well as to induced representations for Lie algebras if we take $R, S$ to be universal enveloping algebras.

I'm not aware of a (useful) free-forgetful interpretation of the tensor-hom adjunction in general.

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Re: the idea of thinking of adjunctions in terms of "optimization," actually I do not recommend thinking in these terms. That part of the Wikipedia article was written by a single person over a decade ago (who happens to be an old friend of mine, incidentally) and does not exactly reflect the practice of a community of category theorists or anything like that. There are enough different ways of thinking about adjunctions that you can take your pick from the available options; I think the one closest to the "optimization" idea is to think in terms of representability.

Namely, every functor $F : C \to D$ poses a "representability problem": are the presheaves $\text{Hom}_D(F(-), d) : C^{op} \to \text{Set}$ representable? $F$ has a right adjoint $G$ precisely iff the answer is "yes" for every $d$, in which case the representing object is $G(d)$. (In particular, the existence of these representing objects alone uniquely determines $G$ as a functor; this is a nice exercise if you haven't done it already.) These representability problems sometimes resemble optimization problems (e.g. in the case of posets, as in your floor / ceiling example) but not always and thinking in terms of representability is fully general; then you can try to build intuition over time for what it really means to ask that a functor be representable. Dually for left adjoints, of course.

I already don't think it quite makes sense to think of, say, the free group functor as the "most efficient way to turn a set into a group." Let's consider a variation where we replace groups with magmas, which are sets $M$ equipped with a binary operation $M \times M \to M$ satisfying no axioms. Here is a very efficient way to turn any set $X$ into a magma: adjoin a new element $0$, and define every product to have value $0$. Easy! However, this is not the free magma on $X$, which is much more interesting; the free magma is the set of (rooted, planar) binary trees whose leaves are labeled by elements of $X$.

I don't think "efficiency" is the right word to describe what makes these constructions different. Here I think it's worth taking "freeness" seriously as a word: the free group (or free magma, etc.) is the freest way to turn a set into a group (or magma, etc.) in the sense that it imposes the fewest constraints on which elements of the group are equal (only the constraints imposed by the axioms). Representability, as it turns out, is exactly how to make this idea precise.

That is, let $U : \text{Grp} \to \text{Set}$ be the forgetful functor from groups to sets. What does it mean to ask that the left adjoint $F : \text{Set} \to \text{Grp}$ (the free group functor) exist? It means that we want all the functors $\text{Hom}_{\text{Set}}(X, U(G)) \cong G^X$ to be representable, as functors of the group variable $G$, for every set $X$. The representing objects $F(X)$ must satisfy

$$\text{Hom}_{\text{Grp}}(F(X), G) \cong \text{Hom}_{\text{Set}}(X, U(G)) \cong G^X$$

and what this means is exactly that $F(X)$ is a group containing elements labeled by $x \in X$ which are as "free" or unconstrained as possible, in the sense that they can map onto arbitrary elements of any other group $G$. (The map $X \to F(X)$, or more precisely $X \to U(F(X))$, is of course the unit of the adjunction.)

Answer 2

I hope I can add to this discussion. Recently, I was on the MathOverflow page you referenced, and I was coming up with comments that relate both to the "risky/conservative approximations" perspective and the "optimal solutions" perspective. Though my comments all have to do with the extension/restriction/coextension of scalars adjunctions, I believe the general "idea" behind how these adjunctions relate to the various perspectives can easily put the tensor-hom adjunction into perspective (and other adjunctions as well), and I'll try to say a little about how as well.

Risky/conservative approximations perspective

The first comment that I posted was about the "risky/conservative approximations" perspective of left and right adjoints:

I know this is late, but I found the intuition of the right adjoint as "conservative" and the left adjoint as "liberal" useful, even in categories that aren't totally ordered. Let $\varphi: B \to A$ be (commutative) ring map. Then there is a natural functor $\text{Mod}_A \to \text{Mod}_B$ given by restriction of scalars $M \mapsto M_B$. It also has left and right adjoints, given by extension $N \mapsto A \otimes_B N$ and coextension $N \mapsto \text{Hom}_B(A,N)$ respectively. Extension is "liberal", because it must represent all maps $N \to M_B$ going into $M_B$ as a map into $M$, so it approximates $N$ as the "bigger" $A$-module $A \otimes_B N$. Conversely, $\text{Hom}_B(A,N)$ approximates $N$ by trying to represent maps going out of $M_B$ as maps out of $M$, so it has to be "smaller" and more "conservative". Indeed, every element in $N$ can be represented in $A \otimes_B N$ via $n \mapsto 1 \otimes n$ but not vice versa. Similarly, every element $f \in \text{Hom}_B(A,N)$ gives rise to an element $f(1) \in N$, but the reverse need not hold.

For the tensor-hom adjunction, the intuition is as follows: given an $A$-module $P$, we want to construct an $A$-module $F(P)$ that is large enough to represent every map $P \to \text{Hom}_A(N,M)$ as a map $F(P) \to M$. Since $\text{Hom}(N,M)$ is "smaller" in some sense than $M$, the collection of maps going into $\text{Hom}_A(N,M)$ is correspondingly larger. Thus, we require a more "liberal" approximation of $P$ to accommodate these extra maps. This explains intuitively why the left adjoint $P \otimes_A N$ is "bigger" than $P$.

Conversely, if we start from a map $P \otimes_A N \to M$, we want to reinterpret this as a map from $P$ into some $G(M)$. Here, we seek an approximation of $M$ capable of capturing exactly those extra maps obtained from expanding $P$ to $P \otimes_A N$. To capture these extra maps, we approximate $M$ as something "smaller", hence the more conservative approximation of $\text{Hom}_A(N,M)$.

Similar lines of thought explain why there is often the pattern of left adjoints "adding" structure and right adjoints "taking away" structure.

EDIT:

I realize now that "bigger" and "smaller" were the incorrect words to use. First of all, the tensor product of two modules can end up with smaller cardinality due to torsion. For example, $ \mathbb{Z}_3 \otimes \mathbb{Z}_4 = (0) $, so "bigger" is a misleading term. Note also that if $ V, W $ are finite-dimensional vector spaces of dimensions $ n, m $, then $ \dim(\text{Hom}(V, W)) = nm $, so "smaller" is also misleading.

But my thinking at the time of writing the above was this: If I have $ p \in P $, then for any $ n \in N $, I have the natural map $ (p, n) \mapsto p \otimes n $. But if I have $ n \in N $ and $ m \in M $, there is not necessarily a way to find a map $ \phi \in \text{Hom}_A(N, M) $ such that $ \phi(n) = m $ (such a map may not exist).

So I called the former "bigger" because I could associate to each pair $ (p, n) $ a canonical element in $ P \otimes_A N $, and I called the latter "smaller" because I could not necessarily associate to a pair $ (n, m) $ a natural element in $ \text{Hom}(N, M) $.

But note that for a map $ f: P \to \text{Hom}_A(N, M) $ going into $ \text{Hom}_A(N, M) $, I can associate to a pair $ (p, n) $ a natural value in $ M $, namely $ f(p)(n) $. Similarly, a map $ \tilde{f}: P \otimes_A N \to M $ going out of $ P \otimes_A N $ also gives me a natural way to associate a pair $ (p, n) $ with a value in $ M $, namely $ \tilde{f}(p \otimes n) $.

So these approximations are on equal footing when we add the third module $ P $ or $ M $ into our picture, and consider maps going in or out of them. Both give me a way to associate $ (p, n) $ with a value $ m $, where $ p, n $ are always treated as inputs, and $ m $ is always treated as an output.

The "bigger" quality I was thinking of with respect to $ P \otimes_A N $ and the ability to associate every pair $ (p, n) \mapsto p \otimes n $ is simply a consequence of needing to treat the pair $ (p, n) $ as an input to be mapped out. So there has to be an associated object for every pair $ (p, n) $ in $ F(P) $ — i.e., a more "liberal" approximation.

The "smaller" quality of not being able to associate to a pair $ (n, m) $ a natural thing in $ \text{Hom}(N, M) $ comes from the fact that for a pair $ (n, m) $, $ m $ would have to be an output to be mapped into — we want $ \phi(n) = m $ in $ \text{Hom}(N, M) $. This way, we can have $ f(p) = \phi $ and $ f(p)(n) = m $. (Note the lack of restriction on $ n $, since $ N $ is the domain of the map $ f(p) $.) So this restricts the possibilities for what $ m $ can be — i.e., a more "conservative" approximation. (A way to think about it is that the Hom approximation simply has to record the possible outputs $ m $ for each $ n $, while the tensor approximation has to create a possible input $ p \otimes n $ for each $ p $ and $ n $.)

That these two say the same thing for triples $ (p, n, m) $ is the precise property of adjoint functors: $ p \otimes n \mapsto m $ carries the same information as $ p \mapsto [n \mapsto m] $.

In general, the functors are really telling you the best approximations of something in the context of compatible morphisms going in (right adjoints) vs. compatible morphisms going out (left adjoints), so that for the former, they are approximated as "outputs" and for the latter, as "inputs". Their compatibility has something to do with the input–output pair carrying the same information. (In fact, one can see this input or output nature in the units and counits, respectively.)

My comment above should probably be revised too, but oh well… Hopefully this clears things up. I'm still trying to understand things myself.

An adjoint pair gives you a pair of approximations between two categories that are "compatible" with each other. This "compatibility" is explored more in the optimization perspective.

Optimization perspective

While I think Qiaochu is right that this is a more "particular" way of looking at things, I believe this perspective can be made precise and general. I came to this realization while trying to answer the following comment under Andrew's answer (which you also reference):

In which sense a functor poses a question? For example, what is the problem posed by the functor $M \rightarrow M \otimes_A B$?

The answer I came up with (that was too long to post there, so I'll post it here) was this:

I hope I'm understanding the interpretation given by Andrew correctly. To answer your question about the extension of scalars functor, recall that the functor $M \to M \otimes_A B$ is left adjoint to the restriction of scalars functor $M \to M_A$. Thus, the "question" is really posed by the restriction of scalars functor: it asks, "Given an $A$-module $N$, how can I approximate $N$ as a $B$-module in a way that is compatible with the functor $M \to M_A$?" Precisely, fix an $A$-module $N$ and consider the category $\mathbf{C}_N$ whose objects are $A$-module morphisms $N \to M_A$, where $M$ ranges over $B$-modules, and whose morphisms are commuting triangles $f_A: (N \to M_A) \to (N \to P_A)$, formed by scalar restricting $B$-module morphisms $f: M \to P$. We say that a $B$-module $S$ is an "approximation of $N$ compatible with $M \mapsto M_A$" if the induced $A$-module map $N \to S_A$ is weakly initial in $\mathbf{C}_N$ (i.e., for every object in $\mathbf{C}_N$ there exists at least one morphism from $N \to S_A$). If there is an approximation (of $N$ compatible with $M \mapsto M_A$) $S'$ that is universal with respect to this property (so that $N \to S'_A$ is initial in $\mathbf{C}_N$), we call $S'$ an "optimal approximation of $N$ compatible with $M \mapsto M_A$" and call the initial map $\eta_N: N \to S'_A$ the "unit" of $N$. If such an optimal approximation exists for every $N$ in $\text{Mod}_A$, then the assignment $N \mapsto S'$ defines a functor (since the universal property ensures that morphisms $N \to N'$ induce unique maps between the corresponding optimal approximations). This functor is precisely the left adjoint to the restriction of scalars functor. One can then show that the extension of scalars functor $N \mapsto N \otimes_A B$ satisfies the property of being an optimal solution for $\mathbf{C}_N$. In particular, the unit $\eta_N: N \to (N \otimes_A B)_A$ can be given explicitly by $n \mapsto n \otimes 1_B$. This exhibits $N \otimes_A B$ as the optimal approximation with respect to $M \to M_B$ of $N$ as a $B$-module, thus (optimally) answering the question posed above.

I do think Qiaochu's representability perspective might be related to the one I came up with above, but I don't understand his perspective enough to confirm this (I don't know what representable means).

Furthermore, note that I tried to phrase my answer in a way that would make it obvious how to generate the "left adjoint question" posed by any functor $G$ and what it means for the left adjoint $F$ to "give a solution". It should also be possible to phrase things the other way with a "right adjoint question" and a corresponding solution.

Note that I made up some of the terminology in an effort to help make this perspective precise, so if there are already standard names for some of these things, please let me know.

EDIT: I'm not certain about this, but I was told that what I have described here is essentially the adjoint functor theorem, though I haven't put in the work to really investigate this yet.