ANSWERS TO QUESTIONS ASKED BY STUDENTS in Math 245, taught from my notes "An Invitation to General Algebra and Universal Constructions", /~gbergman/245, Fall 2015. ---------------------------------------------------------------------- Regarding the paragraph preceding Proposition 3.4.6, you ask how the description of what s_v does to a implies that s \neq t in T_{red} implies s_v(a)\neq t_v(a). Well, the description shows that if s\in T_{red}, then s_v takes a to the word consisting of precisely the string of symbols of s, with parentheses removed, followed by a. Now if s and t are different members of T, then on removing parentheses, they still have different strings of symbols, since members of T_{red} all have parentheses clustered to the right (see discussion leading to (3.4.1)) and "^{-1}" applied only to elements of X, not to longer parenthesized expressions. Hence adding an a at the end, we get different words s_v(a) and t_v(a). Does this make sense now? ---------------------------------------------------------------------- You ask about the "striking properties" of the group introduced in Exercise 4.3:12. The most striking one involves the concept of a "left orderable" group -- a group G that can be given a total ordering "\geq" such that whenever two elements g,h\in G satisfy g\geq h, and f is any element of G, then one also has fg\geq fh. It is easy to show that such a group has no element of finite order other than e, and for many years it was an open question whether the converse was true. This group turned out to be a counterexample. (I've recently learned that the person who discovered this fact, David Promislow, had been running a computer program to test whether the group had a certain property known to be necessary for a group to be right orderable. The computer turned up a counterexample to that property, and Promislow at first thought there must be an error in his program; but he checked it and found it was right.) ---------------------------------------------------------------------- > Why is the commutator subgroup also called the derived subgroup? I don't know. It might simply be that early in the development of group theory, it was the one important general way of constructing a subgroup from a group, so it was given a very "basic" sort of name. Or it might be that group theorists first started using the symbol G' for this important construction, and then gave it the name "derived" because in calculus, f' denotes the "derivative" of f. ---------------------------------------------------------------------- You asked whether the map X --> F has to be an inclusion. Please remember, in submitting future Questions of the Day, to specify what point in the text your question refers to. I think you were referring to the map from a set X to the underlying set of the free monoid on X. In constructions of free algebraic objects, the map u: X --> F does not have to be an inclusion. For instance, one of the constructions of a free abelian group that we saw in section 4.4 took the underlying set of the group to consist of integer-valued functions on the set X, and the universal map carried each x\in X to the function X --> Z which had the value 1 at x, and 0 at all other points. The element x is certainly not the same as that function. The map u will, however, in general be a one-to-one map (often called an "injection".) In our notation, we often use the same symbol, e.g., x_i, for an element of x, and its image in our universal object, relying for context to tell us when x_i itself is meant, and when u(x_i) is meant. But that is simply a shorthand to avoid messy notation. ---------------------------------------------------------------------- You ask whether for monoids, as for groups, the free object on a larger number of generators can be embedded in the free object on a smaller number of generators. Yes. Actually, for monoids examples like that are "easier" to come by in one way, but "harder" in another, than for groups. "Easier" in that within the free monoid on \{x,y\}, the countably many elements xy^n (n=0,1,2,...) are free generators of a free submonoid. This is not so for groups: if we abbreviate xy^n to z_n, we find that (z_m)^{-1} z_n = (z_0)^{-1} z_{n-m}, so the elements z_0, z_1, ... satisfy nontrivial group relations. (However, there is also an easy example in free groups: the elements w_n = y^{-n} x y^n generate a free subgroup.) "Harder" because while it is known (though not that easy to prove) that every subgroup of a free group is free on some set of generators, this is not true in monoids. E.g., if x\in X and F is the free monoid on X, then the elements x^m as m ranges over all nonnegative integers other than 1 (or more generally, over all nonnegative integers greater than some fixed m>0), form a submonoid which is not free. ---------------------------------------------------------------------- You ask about the 1/3 and 2/3 in Fredman's Conjecture. These are forced on us by the 3-element partially ordered set that has one pair of comparable elements, and one element incomparable to both of these. (E.g., the set of integers {2,3,4} under divisibility.) There are only three linearizations of that ordering, so for any pair of incomparable elements, the ratio of the linearizations that put one above the other to those that put them in the reverse order can only be 1:2 or 2:1; so the former have to constitute 1/3 or 2/3 of the linearizations. One can get more examples where the best one can do is 1/3 or 2/3 from the above one; e.g., by throwing in a chain of elements all lying above the three elements mentioned, and/or a chain of elements all lying below them, or in more ingenious ways, such as by putting one copy of the above poset on top of another. But I would guess that if one excluded posets constructed in these ways, then on those that remained, one could assert some narrower interval around 1/2 than [1/3, 2/3]. (One might do this exclusion by defining on each poset P the equivalence relation generated by the condition of being incomparable under the given ordering. Then the posets that would have to be excluded would be those in which all equivalence classes had cardinality 1 or 3, and the 3-element equivalence classes had two elements comparable under the ordering.) ---------------------------------------------------------------------- You ask whether there are systems of set-theoretic axioms that people use other than ZFC and ZF. There are certainly axioms that can be thrown in in addition to those of ZFC, to yield "stronger" set theories, which people find interesting. We will adopt one such axiom, the Axiom of Universes, in section 7.4. There are many others that set theorists look at. As for axiom-systems that don't include the axioms of ZFC, I haven't heard of any of these being used in a big way; but one that people find interesting, and perhaps as "reasonable" as ZFC, is ZF + the Axiom of Determinacy. The Axiom of Determinacy concerns 2-person games. Suppose one has a 2-person game in which the players move alternatively, each player knows what moves the other has made, and a winner is always determined (via rules that the players know) by the sequence of moves. If the game always ends after a fixed finite number of moves, then it is not hard to prove that there is a strategy that guarantees one or the other player a win. On the other hand, my understanding is that from the Axiom of Choice it can be proved that there exist infinite games (where the players make alternate moves for countably many rounds, and the winner is determined by the infinite sequence of moves) in which for every strategy that one player can adopt, there is a strategy for the other player that will defeat it. However, it seems that this cannot be proved without the Axiom of Choice; and in fact, the Axiom of Determinacy, which says that for every such a game there exists a winning strategy for one of the players, appears to be consistent with ZF, so that ZF+Determinacy is a plausible alternative to ZFC. I just know about the subject from hearsay. There is a Wikipedia page about such infinite games, https://en.wikipedia.org/wiki/Determinacy . If you're interested in following this up, you should ask one of our set theorists about it. ---------------------------------------------------------------------- In connection with the statement (three paragraphs after display (5.4.1)) that there is no easy way to extend the definition of an ordered pair as the set \{\{X\},\{X,Y\}\} to a definition of ordered n-tuple for larger n, you ask why we can't just define the ordered n-tuple (x_1,...,x_n) recursively as the ordered pair ((x_1,...,x_{n-1}),x_n) when n > 2. Nice! I hadn't thought of that. However, there are still a couple of advantages in to the approach described in the text. On the one hand, it has the useful property that if an m=tuple (x_1,...,x_m) and an n-tuple (y_1,...,y_n) are equal, then m=n and x_i=y_i for all i; while the approach you suggested would have every n-tuple also being an m-tuple for all m How do we formalize the notion of a construction that may not > necessarily be a function within ZFC? ... We formalize it as a rule which describes, in every case, what the value should be. This is like the situation to which the Axiom of Replacement is applied. > If the set of functions F is fixed, it seems like we should be able > to treat r as a function from a subset of X x F -> {r(x, f_{ Why 0 and 1 for the notation of the least upper bound and the > greatest lower bound of the empty family rather than some other > notation (possibly involving $\infty$)? In the lattice of subsets of a set, looked at as \{0,1\}-valued functions, the empty set is the constant function 0, and the total set is the constant function 1. Also, just as in rings, 0 and 1 are the neutral elements for + and ., so in lattices, a least and a greatest element will be neutral elements for \vee and \wedge. ---------------------------------------------------------------------- Regarding section 5.7, you write, > ... You say anything proved within our systems may model > the real world ... What I meant is that if we set up a mathematical model of some aspect of the real world, say in terms of differential equations, and we ask a question about how that model behaves, and answer it with the help of the Axiom of Choice, then assuming the question is equivalent to the question of how some numerical computations come out (say computations that approximate the differential equation more and more closely using finite-difference equations), then what we deduce using the Axiom of Choice must be consistent with the results of our computations, and so must represent the behavior of the real world with as much accuracy as our model does. > My question is do you know any examples where it is "convenient" to > accept AC ... I think the statement that every vector space has a basis is such a result -- it allows us to picture "exactly" what all linear maps between two vector spaces look like. For instance, we can say that given vector spaces V and W, any homomorphism f from a subspace of V into W can be extended to a homomorphism from all of V into W. The existence of such extending homomorphisms may themselves not be useful facts about the real world, but it shows us that knowing that such a homomorphism f can be extended, there is no point in looking for restrictions that this implies on f; and it is useful to know what not to waste our time on. (In contrast, the statement that, say, an additive homomorphism of abelian groups f: 2\Z --> \Z can be extended to all of \Z does restrict f: such an f must have image consisting of even integers, though a general map 2\Z --> \Z need not.) > ... and also do you think it would be "inconvenient" to work > in a system either without AC or one with the negation of AC? Yes. ---------------------------------------------------------------------- > In Definition 6.3.2 you say the "class of subsets". What does > this mean? It means "set of subsets". In general, "class" has a wider meaning than "set", as discussed in the next-to-last paragraph of section 5.4 (i.e., the next-to-last of the paragraphs preceding Exercise 5.4:1). But as noted in the paragraph after that one, it is also used in contexts where the only relevant classes are sets, so in these cases it means the same as "set". ---------------------------------------------------------------------- > In example 6.5.6 what is a radical ideal? In a commutative ring R, the radical of an ideal I is the set of all elements r\in R such that some power r^n lines in I. This is itself an ideal. A "radical ideal" is an ideal that is its own radical; i.e., that contains r if it contains r^n for some positive integer n. In the context of 6.5.6, note that if a polynomial f has the property that f^n(a_1,...,a_n) = 0 for some (a_1,...,a_n)\in \C^n, then f(a_1,...,a_n) = 0 for the same (a_1,...,a_n). Using this observation it is not hard to see that for any subset A of \C^n, the set of polynomials A* is a radical ideal of the ring of polynomials. ---------------------------------------------------------------------- > ... about the etymology of the word involution ... Never thought of that! I looked it up in the OED. It seems that "involution" is a noun from the root of "to involve", and most of the nonmathematical meanings that it gives have the idea of entanglement. It gives three mathematical meanings: An old one, which apparently meant raising a number to a power; a second one, which I was familiar with as a map of the plane into itself which in appropriate polar coordinates has the form (r,\theta) |-> (r^{-1},\theta) (though they only refer to functions of the line into itself), and finally "A function or transformation that is equal to its inverse." The OED is not big on explaining how meanings come from each other. My guess is that "raising to a power" arose from the idea of a number being "entangled with itself", and is unrelated to the other two (though the geometric sense could be somehow related to r^{-1} being a power of r, or, if one reverses the sign of \theta, to taking inverses in the complex plane). I think that the (r,\theta) |-> (r^{-1},\theta) sense might have come from a biological sense that they show, "A rolling, curling, or turning inwards" on the part of an organ. This fits with the etymology of "in+volu-" = "in-turning". Something that "turns inwards" often turned inside out, and if "involution" came to have that meaning, it would easily fit (r,\theta) |-> (r^{-1},\theta). Then this could have been generalized to any function of order 2, giving their final sense; and then specialized within ring theory to an order-two map with the property I state. (By the way, the map (r,\theta) |-> (r^{-1},\theta) has an interesting geometric property: it takes \{lines and circles\} to itself. And on points with rational values of r, it preserves the property of having rational distances from each other (by an easy observation involving similar triangles). So it is a useful tool in studying families of points with rational distances among them.) ---------------------------------------------------------------------- > In Definition 7.1.2, an identity morphism for each object is a > requirement for any category. I've also seen categories defined > more generally and a unital category defined as a category in which > every object has an identity morphism. Why do you choose to go by > this standard rather than the more general one? Anything can be generalized -- one could throw away the associativity of composition, for instance. And such generalizations might be valuable for some special purposes. But the standard definition of category embodies the properties that hold in the vast majority of motivating cases, and all aspects of it are useful. (The "nonunital categories" that you refer to, if one wishes to study them, are easily embeddable as "ideals" in unital categories, so that the theory of unital categories provides tools for studying them.) ---------------------------------------------------------------------- You write that you feel that the Axiom of Universes doesn't seem as believable as the other axioms, which you feel are clearly true. Well, I don't consider the axioms of set theory to be "true" or "false" (cf. section 5.7); I would judge them in terms of whether or not they form a useful model for the way we think about collections of entities, which enables us to reason precisely about these. Regarding the Axiom of Universes, the first few paragraphs of section 7.4 give reasons for setting up a set theory in which universes exist: that if we start with a set theory that merely satisfies ZFC, we would like to be able to talk about the collection of "all sets"; but that won't be a set. If we set up assumptions that that allow us to treat these classes just like sets, why not rename them sets and assume ZFC applies to these new things we are calling sets? And if we do this once, why not allow the process to be iterated indefinitely, and express this as the Axiom of Universes? We can never get away from the problem that *all* the things we are considering sets will be a collection that is not a set; but we'll have a system where the damage that that fact does has been essentially eliminated. Finally, as I said in class (I don't know whether you were already away at that point), most of the Axioms of ZFC consist of weakenings of the Axiom of Abstraction (described about one page after the list of axioms of ZFC). The general rule seems to be that any weakening of that axiom is OK if it doesn't allow us to define a set S in a way that requires one to already "have" S available to consider in applying the criterion for membership in S. And the Axiom of Universes is OK by that standard: each universe is built up from sets constructed "before" it. So -- since the Axiom of Abstraction seems intuitively "almost true" -- it is reasonable to accept this instance of it as "true". > ... Is it 'reasonable' to say that 'small sets' in the 'usual > sense' correspond to elements of some such minimal universe? Well, I'd rather not make such a convention. ZFC doesn't preclude the existence of universes, so if you make the above convention, then things that people just assuming ZFC called "sets" would not necessarily be "small sets in the usual sense" under your convention. Moreover, set theorists like to study "large cardinal axioms", e.g., the existence of a "measurable cardinal", and my understanding is that almost all (all?) large cardinal axioms imply the existence of some inaccessible cardinals, equivalently, of some universes (though not necessarily the Axiom of Universes). ---------------------------------------------------------------------- > You mention the notion of a large group in this exercise. Do > examples of this naturally appear? Are there interesting theorems > about these or are they essentially the same as small groups? Remember that a group that is "large" with respect to one universe will be "small" with respect to another universe. Generally, one will study a given group within a universe to which it belongs, and there it will always be "small". Anyway, it's easy to get groups of arbitrarily large cardinalities; e.g., free groups on big sets; and a group of large cardinality won't lie, even up to isomorphism, in a universe of smaller cardinality. Perhaps your question really means "Are there interesting results that hold for groups lying in some universes that don't hold for groups lying in others?" Well, set-theorists are interested in "large-cardinal axioms", and if we consider a universe determined by a cardinal of the sort that one of those axioms concerns, some set-theoretic statements will be true that are not true in a sub-universe not satisfying the same large-cardinal axiom. Probably there are ways of encoding some of those set-theoretic statements in terms of the existence of groups with specified properties. But I'm not familiar with the field, and can't say whether such results would be of group-theoretic interest. ---------------------------------------------------------------------- > ... Can we make a Third, Fourth, or nth Axiom of Universes? ... Certainly; but I don't see the point. The Axiom of Universes is simple, and does what we wanted and more. There's no reason to assume that the next challenge to the adequacy of set theory, if there is one, will come from the same direction; so rather than barricading ourselves against danger from that direction, we should just be on the lookout for what may come. (If it does come from that direction, we could just add such axioms.) > Another way of saying this would be: how long of a chain of > universes (ordered by "\in or =") can we get? ... Well, as you note, using the Axiom of Universes, one can get chains of universes as large as any ordinal (in any of those universes)! > ... the following reasonable-sounding axiom: "given any collection > of sets, there is a universe containing all of them," ... That won't work: "all sets" is a "collection" of sets, but since a universe is itself a set, we can't have a universe containing them all. ---------------------------------------------------------------------- Concerning my comment in the third paragraph of section 7.4 that "one could remove the assumption that every member of a class must be a set, so as to allow certain classes of proper classes, and extend the axioms to apply to such classes as well as sets..." you write "In this case, we still have that those proper classes are classes of sets, right?" No. For instance, if A is the class of all cardinals, and B the class of all ordinals, and we allow classes of such proper classes, then \{A,B\} will be such a class; but it will not be a class of sets, since its members, A and B, are not sets. (Rather, it will be a class of classes of sets.) > So that later when we rename them large sets and small sets, ... I hope you understand that what we do after Definition 7.4.1 is relative: we say that a set is "small" or "large" relative to a choice of universe. For most later purposes, it will suffice to leave this choice in the background; but for understanding the concept, you should think "small with respect to U" and "large with respect to U" rather than "small" and "large". > ... we have that the collection of all small sets form a large > set, but the collection of all large sets does not. ... Could we > not once again remove the assumption that every member of a class > be a set and extend the axioms to such classes, ... Well, suppose we change the first sentence above to "the collection of all sets in a universe U is a member of the next universe; let's call it U', but the collection of all sets in U' is not a member of U' ..." But assuming the Axiom of Universes, it is a member of the universe after U', say U''. So from that point of view, I think the Axiom of Universes achieves what you want. Of course, it's true that the collection of *all* sets in our new set-theory is not itself a set in that theory. But that will be true in any set theory that avoids Russell's paradox. ---------------------------------------------------------------------- > It seems like antihomomorphisms of groups > should have a similar role in group theory > as contravariant functors do in category theory, > so why are these rarely talked about? Every group has a canonical antiautomorphism, the map x |-> x^{-1}; so to give an antihomomorphism a: G --> H is equivalent to giving a homomorphism, f: G --> H defined by f(x) = a(x)^{-1}. So anything one might want to express in terms of antihomomorphisms can be expressed in terms of homomorphisms. Monoids and rings don't have such canonical antiautomorphisms, so one does, occasionally, look at antihomomorphisms among them. In particular, one often talks about involutions of rings (see 2nd half of paragraph before (6.4.1)). Aside from this, though, on those occasions when antihomomorphisms of rings R --> S come up, one most often writes them as homomorphisms R^{op} --> S; I guess because, as things that don't come up often, it is more comfortable to describe them in terms of things one uses regularly. On the other hand, contravariant functors are very common in mathematics, so one refers to them as such. ---------------------------------------------------------------------- > ... if we have a faithful functor from X to Y and a faithful functor > from Y to X, (and, say, both are injective on the object-sets) > are they ismorphic? Nope. There are counterexamples with 1-object categories X and Y, where the monoids of endomorphisms of the unique objects are in both cases abelian groups. (Not finite abelian groups, of course.) Can you find such an example? ---------------------------------------------------------------------- > ... Is there an accessible paper which gives precise criteria for > epimorphisms in the category of rings? Yes. See John Isbell's series of papers, Epimorphisms and dominions. 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 232-246, Springer Epimorphisms and dominions. II. J. Algebra 6 (1967 7)-21 Epimorphisms and dominions. III. Amer. J. Math. 90 (1968) 1025-1030 Epimorphisms and dominions. IV. J. London Math.Soc. (2) 1 (1969) 265-273 Epimorphisms and dominions, V. Algebra Universalis 3 (1973), 318-320 But (as Isbell was always concerned with pointing out), the statement of the criterion, his "Zigzag Lemma", in the first of the above papers is wrong; it is corrected in paper IV. > Also, are the problems in characterizing epimorphisms for rings > similar to those for monoids? Yes. ---------------------------------------------------------------------- > It seems that the most intuitive definition of equivalence of two > categories is that they have isomorphic skeletons. Why didn't you > present this one first? Well, the definition I gave tends to be the one that comes up more in the motivating situations -- we have a way of constructing an object of D from any object of C, and vice versa; and while the composites of those constructions are not quite the identity functors of C and D, there are obvious isomorphisms of them with those identity functors. On the other hand, the skeleton of a category, while formally convenient, can be intuitively rather far from the original category -- e.g., in a skeleton of Group, we can't look at the various infinite cyclic subgroups of an infinite cyclic group Z as distinct groups -- they're "the same" group Z, mapped into itself by different morphisms. Anyway, as I've said before, it's good to have different ways of understanding the same mathematical concept. Which one gets introduced first is often a lesser matter. ---------------------------------------------------------------------- > ... you allude to a nontrivial morphism between morphisms between > morphisms in the category CatCat. ... Not to a nontrivial morphism, but to a nontrivial *concept* of morphism; i.e., a way of defining "morphism between morphisms between morphisms" that doesn't just reduce to something one can define in any category or Cat-category. Recall that in defining the concept of "morphism of functors", we used the fact that functors have objects as their outputs, and that there can be morphisms from the objects produced by one functor and the objects produced by another. Now if we have two morphisms P and Q between a pair of functors F and G between Cat-categories C and D, the morphisms that comprise P and Q may in turn have morphisms between them (because D is a Cat-category), and if some choice of these morphisms give the proper commuting diagrams, then we consider the resulting family of morphisms to be a morphism m from P to Q, i.e., a morphism m of morphisms P, Q of morphisms F, G between objects C and D of CatCat. ---------------------------------------------------------------------- > Another example of an enriched category: the hom-sets in the category > the hom-sets in the category of sets with relations as morphisms have > a boolean algebra structure. That's an interesting observation. But to make it an enriched category structure, one would have to figure out how composition of morphisms behaves on the given pair of Boolean algebras. It looks to me as though it will respect joins, but not meets or complements; so maybe it has to be weakened to an upper semilattice structure. ---------------------------------------------------------------------- You ask whether there is a connection between units and counits of adjunctions, and units and counits of algebras and coalgebras. Well, given an adjunction, if we write UF = T, then the unit gives us a morphism I --> T, and from the counit we can get a morphism T^2 --> T. In general, a functor T from a category into itself given with morphisms I --> T, T^2 --> T satisfying certain conditions is called a "monad" by some, a "triple" by others; and it is thought of as analogous to a monoid M, looked at as a set |M| with maps |M|^0 --> |M|, |M|^2 --> |M| satisfying similar identities. Since the map |M|^0 --> |M| is called the unit of M, the corresponding operation of a monad is called the unit of the monad. A dual sort of structure, called a "comonad", has a "counit". Adjunctions, being very symmetric, have both. The above connects the unit of an adjunction with the unit of a monoid. If one abstracts the definition of a monoid in terms of the category Set, then applying the same concept to the enriched category Ab, one gets the concept of a ring; or, using instead of Ab the category of k-modules, the concept of k-algebra, which you asked about. Dualizing, one gets the concept of coalgebra. I'm not sure whether one can actually bring the concepts of monoids and monads under one hat; I suspect so, namely that if M is a monoid, then the functor |M| \times -- : Set --> Set becomes a monad. Likewise for rings, replacing \times with \otimes: Ab --> Ab. ---------------------------------------------------------------------- You ask whether, if a functor F and a subfunctor G of F both have a limit, the limit of G will be a subobject of the limit of F. This works for Set-valued functors, with "subobject" understood to mean "subset"; hence it works for categories of algebraic objects where limits can be constructed using limits of underlying sets. However, using different choices of what to call subobject, it can fail. E.g., if in Group we choose to define a "subobject" to be a subgroup of finite index, then it fails for infinite direct products. ---------------------------------------------------------------------- You ask about completions of categories. There are various ways of completing a category C, depending on whether you want those limits that previously existed to remain limits, or whether you want your result to be universal among categories given with functors F from C and having limits, even if those limits that C may already have had are not preserved by F. The simplest way to complete C is to take the closure under limits of the image of C under the Yoneda embedding C --> Set^{C^{op}}. Cf. http://ncatlab.org/nlab/show/completion http://mathoverflow.net/questions/59291/completion-of-a-category ---------------------------------------------------------------------- > ... what other important categories have this nice property > that directed colimits commute with finite limits? Because finitary algebras define their operations using finite direct products (a finite limit construction), we will be able to prove in the next chapter, when we study the general concept of algebra, that direct limits of finitary algebras arise from direct limits of their underlying sets. That will be our main use of this result; but that fact can then be used to prove that the result you ask about holds for any variety of finitary algebras. ---------------------------------------------------------------------- You ask about the background of the term "solution-set condition". Well, if you want to describe, say, the group presented by a set X of generators and a set R of relations, then the functor that you want to represent is the one associating to every group the set of X-tuples of its elements which are solutions to the system of equations R. Moreover, the condition that is needed to prove that such a representing object exists is, in classical language, that a set of groups with such X-tuples exists that is as good as the class of all such groups and tuples. (In our language, for "set" and "class" read "small set" and "large set".) The above is my guess as to the background of the term. Unfortunately, mathematicians seldom say what leads them to choose a their terminology. It could, rather, be more like what you suggested: a (small) set that is a "solution" to the requirements of the proof. ---------------------------------------------------------------------- You ask why objects characterized by right universal properties are usually easier to construct directly than those characterized by left universal properties. I think it is because the objects we look at in algebra are defined using maps on direct products of underlying sets, and direct products respect right universal constructions. As a result, right universal constructions on algebras turn out to be based on the corresponding constructions on their underlying sets. ---------------------------------------------------------------------- > ... Is there a way to regard a ring as a one-object category, much > as there is a way to consider monoids and groups? Yes. Rings are 1-object Ab-categories. (Recall that an Ab-category is a category whose morphism-sets all have structures of abelian groups, such that composition is bilinear.) ---------------------------------------------------------------------- > ... what does compact as an element of the lattice of subalgebras > of A mean? Did you look up "compact" in the index? If, after doing so, you still have a question about it, let me know and I'll try to help. ---------------------------------------------------------------------- > What is the point of not assuming disjoint Hom-sets in the definition > of a category? As I say in section 7.3, I don't make that assumption "largely because it would increase the gap between our category theory and ordinary mathematical usage"; since under conventional definitions, where a map f: X --> Y is a subset of X\times Y, f doesn't uniquely determine Y. Actually, in a many-sorted algebra A with family of underlying sets (|A|_i)_{i\in I}, one generally doesn't assume the |A|_i are disjoint; otherwise every time one constructed such an algebra, one would have to check whether one's construction accidentally produce elements belonging to more than one |A|_i. So in studying such algebras, one doesn't want to put in such a requirement. (E.g., in studying actions of groups on sets, one might want to consider pairs G, S where G is a group and S a G-set; these would have two underlying sets |G| and |S|. But sometimes one wants to consider the action of a group on itself.) And a category can be looked at as a many-sorted algebra. So one might say that the choice is between optimizing things for the person who, given that C is a category, wants to say things about it, and for the person who, given some mathematical situation, wants to say "such-and-such is a category". For the former, it would be best that hom-sets be discrete; for the latter, that this not be required. ---------------------------------------------------------------------- > Do people study nonassociative algebras other than Lie algebras? Yes, but probably more work is done with Lie algebras than with all other sorts of nonassociative algebras combined. The next largest area is that of Jordan algebras, mentioned near the end of this section. Another is "power-associative" algebras; i.e., algebras whose 1-generator subalgebras are associative; these satisfy identities such as x(xx) = (xx)x. I have a few papers in which results are proved for nonassociative algebras simply because the arguments needed to prove certain things for Lie algebras (which were what my coauthor in the first two of those papers cared about) didn't really require the Lie identities. The results in the last paper in the group have the curious property that they hold for varieties of k-algebras whose identities have certain properties, satisfied in particular by associative, Lie, and Jordan algebras (and many more); but definitely not all such varieties. I can give you copies if you're interested. ---------------------------------------------------------------------- > Question: Is there a way to generalize the construction of an algebra > object to other types of "objects" made from set-based constructions, > such as posets or topological spaces? ... A difficulty in defining a "poset object" in a category is that a binary relation on a set S is a subset of of the product S\times S, and there is no canonical choice for what a "subobject" of an object of a category should be. One can make definitions based on various choices -- taking subobjects to mean domains of monomorphisms, or equalizer objects -- and see whether any of these have nice properties. Or one could see whether one can characterize the "structure" (in an unknown sense) that an object R of a category C needs to have to determine poset structures on all objects C(R,X) in a functorial way. I just tried Googling "partially ordered object" category and the results all seemed to be about toposes or cartesian-closed categories. I haven't studied these, but I know that they are classes of categories that behave much more like Set than most categories do. This suggests that no one has found a good way to generalize the concept of partial ordering to objects of a "typical" category. ---------------------------------------------------------------------- > Question 2: In the example of SL(n), the category C is a variety. > Is this usually the case when Freyd's theorem is used? Within algebra, certainly. There are other categories of algebras with small colimits; for instance, those defined by "Horn sentences", i.e., implications such as "x^2 = e => x = e" in groups. (If we take the implications "x^n = e => x = e" for all positive integers n, we get the category of torsion-free groups.) But varieties are more often studied. Outside of algebra, I don't know. > Question 1: In practice, how difficult is showing that a given > functor is representable in the sense of Definition 10.3.4? For C a variety of algebras, it's usually pretty clear when the hypotheses apply. Showing that a functor is not representable can be trickier; but since we have just seen that representability as an algebra-valued functor is equivalent to representability of the set-valued functor gotten by passing to underlying sets, Proposition 8.10.4 tells us what we should check. Again, outside algebra, I can't say. ---------------------------------------------------------------------- > Why is the natural correspondence between isomorphism classes > as mentioned in thm 10.6.20 contravariant? It goes back to the Yoneda Lemma, and the point discussed in Remark 8.2.8: Because the hom bifunctor of a category C is covariant in one variable and contravariant in the other, the functor taking each object to the *covariant* hom functor that it induces is *contravariant*. In the present chapter, this comes up in the form: the functor taking a coalgebra object to the covariant algebra-valued functor it represents is contravariant; so for given C and V, the category of co-V-algebras in C is equivalent to the *opposite* of Rep(C,V). This was Corollary 10.3.6. Since the category of E-systems is equivalent to the category of co-monoids in Monoid, it is equivalent to the opposite of Rep(Monoid,Monoid). ---------------------------------------------------------------------- You ask whether one has an analog of Cayley's Theorem for objects of an arbitrary variety of algebras. The examples where we had "Cayley's Theorem" type results were all for classes of structures that were defined as models of certain sorts of mathematical phenomena; and in those cases, it turned out that the sorts of structures so defined modeled the phenomena in question sufficiently nicely that any structure fitting our definition could be realized in the way that motivated the concept. Though the majority of the varieties that algebraists study are models of general sorts of phenomena, an arbitrary variety is not required to be motivated in such a way; so there is no natural formulation of a "Cayley's Theorem". Even when a variety is so motivated, the characterization may not be good enough to give a "Cayley's Theorem". The result of this sort for Lie algebras is the Poincare-Birkhoff-Witt theorem, and this shows that every Lie algebra over a field can be represented by commutator brackets in an associative algebra over the same field; but as noted in the last sentence of Exercise 9.7:2, this is not true of Lie algebras over an general commutative rings. ---------------------------------------------------------------------- Regarding the statement following Lemma 10.10.2 that when k has nontrivial automorphisms and/or idempotent elements, the automorphism class group of the variety of k-algebras has a more complicated structure, you note that you see how automorphisms can be used, but not idempotents. If k = k_1 x k_2, then every k-algebra R can be written R_1 x R_2, where R_1 is a k_1-algebra and R_2 is a k_2-algebra. Hence we can construct the functor R_1 x R_2 |-> R_1 x R_2^{op}. These constructions alone give a group isomorphic to the additive group of the Boolean ring of idempotents of k. Combining these with automorphisms of k, which in general permute the idempotents, and hence induce automorphisms of the above group, we get a semidirect product of the two groups. ---------------------------------------------------------------------- > Just to check my understanding: the results we state in 10.12 > about contravariant right adjuctions also work for contravariant left > adjunctions, but we're only stating one version because contravariant > left adjunctions are so rare compared to left adjunctions, right? No. If you look at Definition 8.12.1, you will see that in the case of contravariant right adjunctions, if C is a variety and we insert in place of the dash in (8.12.2) the free object F_C(1) on one generator in C, then the left-hand side gives the underlying set of V(~). Combining with the right-hand side now shows that the resulting contravariant set-valued functor is representable, namely by U(F_C(1)). Hence the contravariant C-valued functor V is representable by a C-algebra structure on the object U(F_C(1)) of D. But if you turn to the contravariant left adjunction case, shown by (8.12.3), if C or D is a variety, there is in general no way of rendering either side of that formula as a description of the underlying set of U or V as a representable functor. (If C or D is the opposite of a variety of algebras, one can get such descriptions; but this just translates us to the covariant-adjunction case.) > Assuming an affirmative answer to the first question: Why are > contravariant left adjunctions so rare? I'll give you a reprint of my paper. ----------------------------------------------------------------------