Why localize spaces with respect to homology?

Question

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into the subcategory of "local" ones that "see all $E$-homology isomorphisms as equivalences". I'm looking for a motivation for this construction that will make sense to a category theorist.

In particular, I'm not looking for a motivation by applications. Homological localization certainly has many applications, but what I'm hoping for is a "philosophical" argument for why one would expect it to be useful, before even having it in hand.

I'm also not looking for an argument that specializes to localization or completion with respect to primes (which can be described as localization with respect to $\mathbb{Z}_{(p)}$-homology and $\mathbb{Z}/p$-homology respectively) and then says something about the behavior on nilpotent spaces. I feel like I have a pretty good motivation for localization and completion of nilpotent spaces, by analogy with the importance of the analogous constructions in algebra and the fact that a space or spectrum can be considered a sort of "generalized algebraic object" put together out of its homotopy groups (e.g. this is well-described in More concise algebraic topology). But homological localization is only one way to generalize these constructions to non-nilpotent spaces; why is it a good one, and why should we think about doing it for other homology theories as well?

The best I've been able to come up with so far is the following fairly obvious remark: "Homology (and cohomology) are easier to compute than homotopy, so it's natural to restrict attention to those aspects of a space that can be detected by some homology theory." But this doesn't satisfy me, because there is another much more straightforward functor which "restricts attention to those aspects of a space that are detected by $E$-homology", namely "$E$-homology". (Or, if you prefer to land somewhere homotopical, the free $E$-module spectrum.) Why should we think of instead formally inverting the $E$-homology isomorphisms?

In particular, are there analogous constructions in ("cowardly old") algebra where we get something useful by formally inverting the maps inverted by some other functor? Note that for localization at primes, formally inverting the maps in question is equivalent to tensoring with the localized base ring, so it doesn't argue for why to use the former rather than the latter — while for completion at primes, the naive completion functors, at least, are not reflections into subcategories at all!

Answer 1

A student of mine asking for a motivation unmotivated by applications? Haven't I taught you anything, Mike? (Joking of course.) However, perhaps one way to avoid talking about future applications is to reflect on implicit past applications and the explicitly stated original motivations. This is not to take away from your answer, Craig, you know I agree completely, but I'll give an answer orthogonal to yours.

We know from long past history that focusing on some generalized theory can drastically simplify life. Perhaps the first and still one of the most persuasive examples is comparison of the solutions of the Hopf invariant one problem by $K$-theory and by mod $2$ cohomology. This, together with the knowledge that $K$-theory captures so much of concrete importance (index theorem, etc) certainly argues for focusing on $K$-theory. The immense calculational power of periodicity phenomena also argues for a setting that throws away anything that is not periodic. The search for understanding periodicity phenomena in the stable homotopy groups of spheres goes back more than half a century, and the expectation that localizations at homology theories would be relevant goes back almost as far.

In fact, Frank Adams explicitly conjectured (Conjecture 4.6) in his 1973 Chicago lecture notes "Localization and completion" that localizations with respect to generalized homology theories exist. Pete Bousfield attended those lectures and constructed the conjectured localizations the following year. (I attended too and I remember Pete tapping me on the shoulder and asking "How does he know it is a set?"). In fact, Adams' notes fail to give a proof only because of set theoretic questions. Zig Fiedorowicz has uploaded a version of those notes on the arXiv (http://arxiv.org/abs/1012.5020) and he has added an epilogue showing that an easy modification of the argument Adams originally had in mind proves the conjecture, and "has thus taken the liberty of upgrading" the statement, so that it appears as Theorem 4.6 in the ArXiv version of the notes. To quote from Zig's foreward ``Thus it can now be seen in retrospect that Adams amazingly succeeded in his project of "constructing localizations and completions without doing a shred of work".

Frank's lectures were in a sense all about his motivation for wanting such localizations. So, philosophically, I might argue that motivations for mathematical developments might best be sought in their historical context, rather than abstractly, even for a category theorist. But a category theorist might like what Frank says right at the start (p. 10) "I want to study functors in homotopy theory with the same formal properties as localization, so I'd better say what those properties are". And his list of axioms is quite satisfactorily categorical.

Answer 2

This is perhaps orthogonal to your desire for an a priori reason to motivate localization. Indeed it's intrinsically a posteriori. I still think it's a good reason to care about localization that is deeper than the fact that the assignment $X \mapsto E_*(X)$ is a somewhat computable invariant.

If one has a category of things (like spaces) and some sort of notion of a structured gadget in that category, as long as your localization functor is reasonable (e.g., monoidal, if your gadget is defined operadically), then the image of any gadget in the localized category will remain the same sort of structured gadget. However, if one relaxes the requirements of your structured gadget so that they are only visible to the eyes of the localized category, one can in principle come up with a notion which admits exotic examples which are not the localization of a standard object.

This is pretty airy; let me explain via two examples. A spectrum $X$ is invertible if there is another spectrum $Y$ with $X \wedge Y \simeq S^0$. The collection of equivalence classes of invertible spectra forms the Picard group of the stable homotopy category, $Pic(S^0)$. As Hopkins-Mahowald-Sadofsky show, it's isomorphic to the integers, where $n$ corresponds to $S^n$ (with smash inverse $S^{-n}$).

This doesn't appear to change in the localization of the stable homotopy category with respect to any cohomology theory $E$, since, as you note, the local category is a subcategory of the original category. However, there is a problem, in that the smash product of two $E$-local spectra need not be $E$-local. So one defines a new monoidal structure on the $E$-local category by localizing after smashing:

$$X \otimes Y := L_E (X \wedge Y)$$

Now, one can ask: what is $Pic(L_E S^0)$, the set of isomorphism classes of invertible objects with respect to this new monoidal structure on the category of $E$-local spectra? Of course, the image of any spectra which are invertible with respect to $\wedge$ (i.e., spheres) are still invertible, so I get a map $Pic(S^0) \to Pic(L_E S^0)$. But generally the target group is much larger and more interesting. For instance, when $E$ is mod $2$ K-theory, many exotic examples can be constructed as Thom spectra over $\mathbb{R} P^\infty$. In general when $E$ is a Morava K-theory, computation of $Pic(L_{K(n)} S^0)$ is difficult, and a very active subject of investigation.

This highlights the fact: the notion of invertibility can be (in fact must be) modified in the $E$-local category, and yields a new and richer notion than the original. The same holds in the second example, $p$-compact groups.

A compact Lie group $G$ is a loop space ($G \simeq \Omega BG$), and its integral homology is a Poincaré duality algebra (and in particular finite rank over $\mathbb{Z}$). One may relax these assumptions in the $E$-local setting. In particular, if we take $E =H\mathbb{F}_p$ to be mod $p$ homology, we can declare a loop space $G$ to be a $p$-compact group if it is $p$-local with $H_*(G, \mathbb{F}_p)$ finite rank over $\mathbb{F}_p$. Again, the $p$-localization of a compact Lie group is an example, but there are many exotic examples which arise. Notably, whenever $n$ is a divisor of $p-1$, the $p$-completion of $S^{2n-1}$ is a $p$-compact group (the Sullivan spheres), despite the fact that only $S^0$, $S^1$, and $S^3$ (and, if you're charitable, $S^7$) are groups or, for that matter, H-spaces.

Lastly, I want to point out that these relaxed definitions are not "techniques in search of a problem." In fact, they create a home for existing constructions. For instance, the usual classification of compact Lie groups in terms of the action of their Weyl group on the lattice of the characters of their maximal torus extends to the $p$-compact case. In particular, when $p$ is odd there is a bijection between $p$-compact groups and $p$-adic reflection groups. So: if you were wondering what is the geometry associated to $p$-adic reflection groups, the answer is: $p$-compact groups.

Similarly, in the Picard group setting, there is a spectral sequence that computes $Pic(L_{K(n)} S^0)$ using classical computations of group cohomology of automorphism groups of Lubin-Tate formal groups, acting on the units in the ring of functions on the Lubin-Tate moduli space. So if you're interested in invertible sheaves on that moduli space, you're also interested in exotic elements of $Pic(L_{K(n)} S^0)$.

Answer 3

In particular, are there analogous constructions in ("cowardly old") algebra where we get something useful by formally inverting the maps inverted by some other functor?

Sure. Localize the category of presheaves on a space with respect to stalkwise isomorphisms. The result is the category of sheaves, which is more interesting than just looking at induced maps on stalks.

Or: localize the category of chain complexes with respect to homology isomorphisms. The result is the derived category, which is more interesting than just looking at induced maps on homology.

The point, I guess, is that applying a functor $F$ is certainly a thing that inverts $F$-isomorphisms, but there's no guarantee that it's the universal such thing. You want the universal such thing because you're a category theorist and universal constructions are intrinsically important to you.

Answer 4

Everyone has said very interesting and useful things, but the closest any of them have come to answering the question I wanted to ask (or at least being the sort of answer I was hoping for) is (my interpretation of) Lennart Meier's comment. So here I will expand that and post it as a cw answer.

In 1-category theory, it's become common in certain circles to classify functors as "forgetting properties", "forgetting structure", or "forgetting more than just structure" based on whether they are fully faithful, faithful, or neither. Moreover, we can extract "the structure or properties forgotten by a functor" by factoring it universally through the corresponding sort of functor.

For instance, the forgetful functor from groups to sets is faithful, so it forgets structure (group structure). If we factor it as an essentially surjective functor followed by a fully faithful one, then we obtain a functor to Set that forgets only properties, namely "the property of admitting some group structure".

As another example, consider the category of partial equivalence relations on $\mathbb{N}$ and computable functions that preserve them. Taking quotients gives a functor to Set which is not even faithful. If we factor it as an essentially surjective and full functor followed by a faithful one, we obtain a functor to Set that forgets structure, which we might call "the structure of being the quotient of a partial equivalence relation on $\mathbb{N}$". This latter category, not the original category of partial equivalence relations, is the usual category called "PER" that people study. We can think of an object of PER as a set together with all the possible structure that accrues from its being presented as the quotient of some partial equivalence relation on $\mathbb{N}$.

However, in higher category theory this gets a bit confused. For instance, the usual $(\infty,1)$-categories that we would think of as "spaces equipped with structure", like the category of $A_\infty$-spaces, do not have a faithful forgetful functor — nor is it even necessarily clear what a "faithful $(\infty,1)$-functor" means. However, they do have the closely related property of being conservative, i.e. reflecting isomorphisms.

In 1-category theory, not every functor that we usually think of as "forgetting structure" is conservative (e.g. topological spaces), but many general sorts of structure (e.g. algebraic and coalgebraic) are conservative. Moreover, a conservative 1-functor that preserves equalizers is automatically faithful. So conservativity is not an unreasonable substitute for faithfulness when talking about "structure" for $(\infty,1)$-categories.

Now the point (which has been made by several people) is that for any functor $E$, localizing at the $E$-equivalences is a factorization just like the (eso+full, faithful) one, where now the two classes are (localization, conservative). So if conservative $(\infty,1)$-functors are the ones that "forget structure", then localization at the $E$-equivalences "extracts the maximum possible structure on the images of $E$".

In particular, if $E$ is a homology theory, we can regard the $E$-localization of $X$ as "the $E$-homology groups of $X$, equipped with all the possible structure that accrues from that". As Lennart pointed out, the $E$-homology groups of a space have more structure than being just a graded abelian group, or even being an $E$-module spectrum. The $E$-localization of a space remembers all of that structure, and more: it remembers all the possible structure we might think of (for a particular $(\infty,1)$-categorical notion of "structure") that can be given to the $E$-homology groups.

As was also pointed out, the classical construction of the $(\infty,1)$-category of $\infty$-groupoids from the model category of topological spaces is also a localizaton, at the weak homotopy equivalences. So we can regard an $\infty$-groupoid as "the homotopy groups of some topological space, equipped with all the possible structure that accrues from that" — e.g. actions of $\pi_1$ on $\pi_n$, Whitehead products, etc., and much more. So from this perspective, an $E$-local space is "put together from $E$-homology groups" in an analogous way to how a homotopy type is "put together from homotopy groups".

Answer 5

Maybe this is too elementary an answer, but in my view, localization is a general phenomenon that you might want to do in any category C. You have a class of maps W and you want to form W^{-1}C, the category obtained by formally inverting the maps in W. This is what we do to form the homotopy category of a model category, this is what we do to localize R-modules at prime ideals (or to localize abelian categories at torsion theories) and this is what we do for Bousfield localizations.

So is the question: why do we want to localize at all?

or is the question: how do we pick W?

If the question is "why do we localize at all" I would say that the answer is to make the category we are studying simpler so that we can learn more about it. Many questions about the original category may depend only on the localized one, and therefore be answerable because the localized category is simpler.

If the question is "how do we pick W" (that is, why do we localize with respect to a homology theory rather than with respect to some other W), then I would say because homological localizations are extremely well-behaved examples of a general phenomenon, so we study these localizations because they are easier. We could localize the stable homotopy category with respect to the map from the point to BP, or with respect to the class of maps X --> BP smash X (the BP-nullification), and I would be very interested in that but have no hope of understanding it.

Your question about why E-homology itself doesn't already do the job is analogous to asking why we p-localize instead of merely tensoring with Z/p. We lose much more information by taking E-homology as compared to localizing with respect to E. An extreme case of this is K(n)-homology versus K(n)-localization.

Answer 6

$\textbf{This is a point of view which means that it is only one side of the story}$. Let me start with Gel'fand theorem. The (opposite) category of compact Hausdorff spaces is equivalent to the category of commutative $\mathbb{C}-^{\star}$algebras. Roughly speaking the functor with associate to any space $X$ its algebra of continuous complex vaulted functions $C^{0}(X,\mathbb{C})$ realizes this equivalence.

Gel'fand: $$compact-Hausdorff-spaces \cong_{op} commutative-\mathbb{C}-^{\star} Algebras. $$ $$ X\rightarrow C^{0}(X,\mathbb{C})$$

This equivalence has to potential generalizations. In one way we can ask to which spaces corresponds general commutative rings (commutative $\mathbb{Z}$-algebras).

the answer was given by Grothendieck. $$ affine-schemes\cong_{op} Commutative-rings$$ $$ X\rightarrow \Gamma (X)$$

The other generalization is the noncommutative $\mathbb{C}-\star$algebras and non commutative spaces. In order to continue this story from homotopical point of view we fix a commutative ring $R$, and for any topological space $X$ we define the cochain complex $C^{\ast}(X,R)$, this is an $ E_{\infty}$-differential graded $R$-algebra. There is two interesting cases when $R= \mathbb{Q}$ and $R= \overline{\mathbb{F}}_{p}$. Sullivan theorem says that the homotopy category the localized category of simply connected spaces of finite ($\mathbf{Top}^{fin}$) type with respect to rational homology theory is equivalent to the category of simply connected differential graded $\mathbb{Q}$-algebras of finite type ($E_{\infty}-dgAlg_{\mathbb{Q}}^{fin}$) . Mandel's Theorem says the same in the $\overline{\mathbb{F}}_{p}$ case.

Sullivan $\infty$-equivalence: $$L_{\mathbb{Q}}\mathbf{Top}^{fin}\simeq_{op} E_{\infty}-dgAlg_{\mathbb{Q}}^{fin}. $$ $$ X\rightarrow C^{\ast}(X,\mathbb{Q})$$ Mandel $\infty$-equivalence: $$L_{\mathbb{F}_{p}}\mathbf{Top}^{fin}\simeq_{op} E_{\infty}-dgAlg_{\overline{\mathbb{F}}_{p}}^{fin}. $$ $$ X\rightarrow C^{\ast}(X,\overline{\mathbb{F}}_{p})$$ PS: I did not state the nilpotent version and as I said at the beginning it is just a side of the story.

Answer 7

I'm not sure exactly what you are after, but here is an elementary discussion, surely well known to you. Rephrasing what Mark Hovey said, the first question you should ask yourself is probably why localize at all? There may be several reasons. Here are some:

1. You have some a priori interest in only studying spaces up to a certain equivalence relation (physical, philosophical, application-based). E.g., you study spaces up to homotopy since many physical properties are preserved by continuous deformation....

2. You know some procedure for (more or less) reconstructing what you are really interested in from the localized pieces. E.g., you have arithmetic squares that (try to) piece localizations together.

3. You are really interested in the unlocalized object, but all the tools at your disposal factors through the localization. So you might as well study the localized objects, to rid yourself of information that your methods will not say anything about anyhow. E.g., you want to compute the integral homotopy groups of spheres, but you consistently screw up your signs when looking at odd primary stuff, so you localize at 2.

Common to them all above are that they are various reasons for ignoring exactly certain specified information. Not more, not less.

Now, all of the above reasons apply to why you may what to study E-localization of a space. Examples are given in previous answers. We have arithmetic squares for various E, some people prefer to work with E only, so why not localize? And, obviously, any physically meaningful quantity of a (7,10)-extended QDW-theory on 2-brains, is an E-invariant, right? ;)

Localizing is obviously different from studying just the E-homology of the space, even if you try to be fancy by upgrading the values to, say, to E-module spectra or one of the zillion other natural choices. These other constructions do not forget exactly what you choose to ignore. Localization does. There is of course the interesting subsequent question of relating say the E-completion (constructed as the Tot of an explicit tower, and often more computable) to E-localization (which has the good abstract properties). But that is a separate story...

By the way, Bill Dwyer wrote a beautiful survey paper "Localizations" a few years ago, with lots of examples http://www3.nd.edu/~wgd/Dvi/Localizations.pdf

Answer 8

There are many good answers here...

I don't think Mike is asking about the localization of the category of spaces, but of localizing spaces as special spaces. (Though I could be wrong, and about the former I could say: it's plainly a cousin of Moore-Postnikov resolutions, once removed in categorification)

And I think the best answer to why this is a naturally desirable thing to do is: it tells us that

everything $E$-homology of spaces tells us about spaces is realized in a retract of the category of spaces.

That is, there is a functor $_E\operatorname{Top}\to\operatorname{Top}$ (that, as will follow, forgets only property), and pointing the other way a functor $\operatorname{Top}\to {_E\operatorname{Top}}$ such that the composite ${_E\operatorname{Top}} \to \operatorname{Top}\to {_E\operatorname{Top}}$ is naturally isomorphic to the identity on $_E\operatorname{Top}$. And then the stuff about $ E| : _E\operatorname{Top} \to \operatorname{Spec}$ or whichever you want it to be, reflecting isomorphism and all that.

Answer 9

There are a lot of answers and this is an old question, but I'm surprised nobody mentioned the following perspective.

Question 1: Why localize spaces with respect to homology?

Answer 1: Don't!

(Except as a technical tool.)

Instead, take an interest in homological localizations of spectra.

I think that this answer aligns with the practice of "most" homotopy theorists -- with notable exceptions! (Some interesting results of Neisendorfer come to mind.) The sense that localizing spaces with respect to homology is a "violent" and borderline "unnatural" operation, which perhaps should be replaced by something else, was one motivation for this question of mine. I think I would say (without being terribly knowledgable in the subject) that for unstable localizations, there's less reason to be more interested in the homological case than in the general case.

So we're left with a different, related question:

Question 2: Why localize spectra with respect to homology?

This one I think admits a more systematic answer, which first requires answering a more basic question:

Question 0: Why study spectra at all?

Of course, there are many good answers to this question, but let's look at one:

Answer 0: Brave new algebra + Derived algebraic geometry!

I think these ideas have become commonplace today, but let's sum them up:

• Waldhausen's philosophy of "brave new algebra" stipulates that we study the sphere spectrum $\mathbb S$, spectra, ring spectra, modules over ring spectra, etc. because they form a world of algebra which is better-behaved, more structured, and more fundamental than their decategorifications living in the "cowardly old" world of the ring $\mathbb Z$, discrete abelian groups, discrete rings, discrete modules, etc.

  In particular, the most fundamental aspects of commutative algebra are decategorifications of analogous, even more fundamental aspects of brave new algebra.

• One of the most fundamental aspects of commutative algebra is that it constitutes the local / affine part of the whole subject of algebraic geometry. Categorifying, brave new algebra constitutes the local / affine part of the subject of derived (or spectral) algebraic geometry.

So from this perspective, we can refine our question:

• The category $Sp$ of spectra is analogous to the category of abelian groups: we have that $Sp = Mod(\mathbb S)$ is the category of modules over the initial $E_\infty$-ring spectrum $\mathbb S$ just as $Ab = Mod(\mathbb Z)$ is the category of modules over the initial ring $\mathbb Z$. Equivalently, $Sp = QCoh(Spec(\mathbb S))$ is the category of quasicoherent sheaves over the terminal spectral scheme $Spec(\mathbb S)$ just as $Ab = QCoh(Spec \mathbb Z)$ is the category of quasicoherent sheaves over the terminal scheme $Spec(\mathbb Z)$.

• If $E \in Sp$ (or more generally $E \in Mod(A)$ for an $E_\infty$ ring spectrum $A$, or $E \in QCoh(X)$ for a spectral scheme $X$), then localizing with respect to $E$-homology means passing to quasicoherent sheaves on the open subscheme $Supp(E) \subseteq X$ where the quasicoherent sheaf $E$ is supported.

Question 2 (bis): Given a category of quasicoherent sheaves $QCoh(X)$, why study it via the supports $Supp(E) \subseteq X$ of its objects $E \in QCoh(X)$?

And here I think we have a question which we can actually answer, in a few ways:

Answer 2:

1. Geometrically, what we are doing is getting a handle on the Zariski-open subschemes of our scheme $X$. This is a fundamental thing to do -- it's hard to imagine doing anything if you can't understand the Zariski topology of $X$.

2. The real question becomes: when studying the Zariski topology of $X$, why should we immediately reach for the categorical description in terms of localizations of $QCoh(X)$? After all, in the non-derived world, we tend to get a handle on the Zariski topology of a scheme much more directly.

   In some sense, the answer is that this is a "historical contingency": 30-40 years ago the machinery of derived algebraic geometry was not in place, but the categorical data of $QCoh(X)$ was something people could get their hands on, so they worked with that. It's much like the situation in noncommutative geometry: when it's unclear what the definition of a "noncommutative scheme" $X$ should be, but at least clear what $QCoh(X)$ should be in certain cases, you just do what you can with the category $QCoh(X)$ and you make progress.

3. One might argue that the Zariski topology is not the most important topology (e.g. perhaps the etale or Nisnevich topologies are more important), and consequently we shouldn't put so much emphasis on localizations homological or otherwise. From this perspective, it's again a historical contingency that the Zariski topology was the easiest to get at with older technology (just as it was in the underived world.)

4. You could (and some do!) turn this situation on its head and argue that the functorial viewpoint on algebraic geometry, where one studies $X$ via $QCoh(X)$, really is more fundamental after all. Then we return to the question: why are homological localizations particularly interesting among all localizations? For this I'd appeal to the fact that they have more structure; e.g. the structure of a recollement.

Finally, I think it goes without saying that the case where $X = Spec(\mathbb S)$ is the terminal scheme is particularly fundamental -- just as studying the open subschemes of $Spec(\mathbb Z)$ (i.e. primes!) is so fundamental to algebraic geometry that it goes without saying.

Answer 10

THIS IS NOT AN ANSWER, rather an additional question

I always wanted to know how the following purely abstract-nonsensical (category-theoretic) constructions fit into the particular setup of stable homotopy.

Any object $E$ of any closed monoidal category $(\mathscr S,S,\bigwedge,[\_,\_])$ determines an adjoint pair of functors $E\bigwedge\_\dashv[E,\_]$ and thus both a monad $[E,E\bigwedge\_]$ and a comonad $E\bigwedge[E,\_]$ on $\mathscr S$. This then gives an adjoint pair between $E\bigwedge[E,\_]$-coalgebras and $[E,E\bigwedge\_]$-algebras and one may repeat ad infinitum to (hopefully) finally get some "$E$-local/stable/complete" category. In really good cases it is well related (although rarely equivalent) to $[E,E]$-modules and $E\bigwedge[E,S]$-comodules, but I do not know any good description in general.

Another version (which I learned from Claudio Hermida years ago) would involve adjunctions between (co)algebras and (co)Kleisli categories, rather than coalgebras and algebras. Since the Kleisli construction provides equivalents to the categories of (co)free (co)algebras, this version might be viewed as sort of approximations to the "no-relations" or "field-like" case where "all (co)algebras are (co)free". And if this cannot be achieved to the full, there might be some calculable invariants of $E$ detecting obstructions to doing it.

NB For this to actually work one in fact needs some amount of (co)equalizers; I wonder if (co)fibres would work in the triangulated setting...

If this construction would relate well to the "real thing" this would provide for the category-theorist part of me a nice motivation. Does it?

PS. There is yet another way to produce algebras/coalgebras via the $\textit{contravariant}$ adjunction $[\_,E]\dashv[\_,E]$ (this works in more general setting of a (not necessarily monoidal) closed category) and I might repeat the same question in this context too.