ANSWERS TO SOME QUESTIONS ASKED BY STUDENTS IN MATH 245, taught
in Fall 2017 from my notes "An Invitation to General Algebra and
Universal Constructions", /~gbergman/245.

These are my responses to questions submitted by students in
the course, the answers to which I did not work into my lecture,
but which I thought might be of interest to more students than
the ones who asked them.

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You ask whether the two downward arrows in the diagram in
Definition 3.1.3 represent the same map  h,  or whether there
is a distinction between the map regarded as a set map and as
a group homomorphism.

They do indeed represent the same map  h.  Since a group homomorphism
is a set-map that satisfies appropriate conditions with respect
to the group operations, the same map can be both a set-map and
a homomorphism.  Its occurrence in the triangle is based on the fact
that it is a set map, and so can be composed with the horizontal arrow
of that triangle, and the result equated with the diagonal arrow,
giving us a triangle of set maps.  Its appearance on the right,
on the other hand, is based on its being a group homomorphism.
It is unique for having the combination of the set-theoretic and
group-theoretic properties shown by these two appearances.

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You ask about the remark after the last diagram in section 3.1,
that since any two free groups on  X  are "essentially" the
same, one often speaks of "the" free group on  X.

Whether it is reasonable to use "the" in such a situation depends on
what one is focusing on.  If one is interested in the group-theoretic
properties of the group, these are the same for any two isomorphic
groups, so one thinks of all free groups on  X  as being "the same
for the purposes of discussion", and it is natural to use "the".
On the other hand, if one is interested in questions such as whether
the set  X  is actually contained in  F,  or mapped into it by a
map  u  which is not an inclusion map, then this will differ for
different realizations of the free group structure, and one would
call each of them "a free group on  X".

Regardless of one's point of view, it is not quite precise to say
"the" free group, since the different objects with that universal
property are not literally the same.  But it is OK to speak a
little imprecisely as long as the speaker and hearer understand
what is actually meant.

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Concerning the proof of Proposition 4.6.5, you say

> I don't quite see what the purpose of constructing this set A is ...

I hope that what I said in class answered your question.  Though
it is easy to see that every element of the universal group generated
by images of  G  and  H  can be written in the form (4.6.4), and to
define operations on such expressions, it is not easy to show that
these operations satisfy the group axioms; equivalently, that the
group identities, together with the relations satisfies by our
generators, don't somehow imply equality among two different
expressions (4.6.4).  But by finding a group of permutations of
a set on which all those relations are satisfied, and in which
different expressions (4.6.4) yield different permutations, we
get this conclusion, and hence have the desired description of
the "coproduct" of the groups G and H.

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You ask whether Lemma 4.17.3 can be proved without the Hausdorffness
condition on  K.

No.  For example, let  K  be any set, given with the topology under
which the closed subsets are  K  itself and its finite subsets.
Then any infinite subset of  K  is dense.  Hence if we let  X  be any
infinite set, we can map it into topological spaces K  constructed
as above so as to have dense image, though there is no upper bound
on the cardinalities of such spaces  K.

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In connection with the paragraph before Definition 5.1.4, which
notes that partially ordered sets are not algebras in the sense
used in this class, you ask "Could we not get around this issue
by concentrating on relations instead of operations?"

Yes; but then objects such as groups, rings, etc. would have to
be described as having certain relations  R  subject to the
condition that for every  x  and  y  there exists a unique  z
such that  (x,y,z)\in R;  and results on free groups, rings, etc.,
groups, rings, etc. presented by generators and relations, and so
on, would all have to be formulated in terms of relations with this
particular property.  So to create a comfortable context for studying
these things, I have chosen to work with algebras, defined by
sets with operations, and to consider more general relational
structures to be nearby relatives which we visit when we need them.

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In connection with the suggestion preceding Exercise 5.6:2, that
if you haven't seen proofs by Zorn's Lemma before you might look
at such proofs in standard graduate algebra texts, or ask your
instructor for some elementary examples, you ask for such examples.

Most basic graduate-level algebra texts have many such proofs, but it
takes some work to find where they are.  One text where one can find
these by an online search is Hungerford's "Algebra".  If you go to
https://books.google.com/books?id=e-YlBQAAQBAJ and type "zorn" in
the box saying "Search inside", you will get 23 results, and turning
to the corresponding pages in a physical copy of the text, you can
find the proofs in question.

However, I suspect you can do *most* of Exercises 5.6:2-5.6:14
without looking at additional examples.  It is unfortunate that I
began that string of exercises with 5.6:2, which, in its present form,
does not make it at all obvious how Zorn's Lemma would be used.  So
look at the next few, and see whether you have better luck.

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You asked about my use of the phrase "(generally infinite)" in the
fourth-from-last paragraph of chapter 6.

The reason I put that phrase in was, of course, that we usually
see the conjunction symbol used to connect two propositions,
or, if several conjunction symbols appear, finitely many; and I
wanted to make clear that the finiteness restriction that is
automatic in such cases was not being assumed there.

I could equally well have expressed this by writing "(possibly
infinite)".  In choosing to say "generally", I was implicitly assuming
that in mathematics, infinite structures (such as the ring of integers,
or the real numbers) are more often of interest, and finite structures
a more special case.  But for mathematicians who specialize in the
study of finite objects (e.g., finite groups, finite lattices, etc.),
the reverse is true.  So there was no absolute justification for my
choice of word.

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You ask about the relation between closed subsets, i.e., sets fixed
under "**", where "*" is the operator in a Galois connection, and
topological closure, such as comes up in Galois theory of infinite
field extensions.

Well, closure operations occur throughout mathematics, as the examples
given in section 6.3 show.  It happens that in one area, topology,
one deals with a closure operation that is simply called "closure".
This is perhaps what led you to look at the word "closed" in that way.

I guess when a closure operator is not finitary, i.e., when the
closure of the union of a chain of subsets can contain more elements
than the union of the closures of the sets in the chain, then the
way these new elements arise is often given by a topological closure.
So when this happens, as in the field extension case you mentioned,
topology will be involved.  I don't know whether every closure
operator on a set can be decomposed somehow into a finitary closure
operator and a topology ... .  But anyway, as Exercise 6.5:3 shows,
every closure operator on a set can be looked at as arising from some
Galois connection, but as Exercise 6.3:17(i) shows, those closure
operators that come from topologies alone are highly restricted.

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You ask, concerning the first sentence of Example 7.5.9, where I refer
to "objects of this sort", whether this means sets with operations
of a given signature; and also whether there are any restrictions on
the language in which the propositions comprising  T  are expressed.

My intent was to be very general.  Rereading that Example, I think
that in the first line, after "Let  S  be a set of mathematical
objects", I ought to add, "of a given sort (e.g., of groups, of
positive integers, of topological spaces, ...)".  Then, hopefully, the
sense of "of this sort" later in the sentence will be clear.  The
propositions can likewise be in any language -- all that is needed
to get a Galois connection is that for each object  s\in S  and each
proposition t\in T,  it makes sense to say whether  s  satisfies  t.
The point of this Example is show that the basic ideas of studying
sets of mathematical objects determined by propositions that they
satisfy, and sets of propositions determined by the mathematical
objects that satisfy them, can be looked at as an example of the
concept of Galois connection.

Of course, when one wants to study such a situation further, one will
generally want restrictions of one sort or another.  For instance,
the paragraph following Exercise 6.3:8 discusses certain properties
that follow if the language in question includes the operators
"or" and "and" (interpreted in the standard way), and  T  is closed
under those operators.  The material of Chapter 9 concerns the
situation you asked about, where  S  consists of all sets with
operations of a given signature.  There,  T  consists of all identities
in those operations (but does not contain expressions formed with
"or".  Whether we allow "and" (written "\wedge") doesn't really
matter, because, e.g., if  t_1,  t_2,  t_3  are identities, then the
set of algebras satisfying all members of, say  \{t_1\wedge t_2, t_3\}
is also the set satisfying all members of \{t_1, t_2, t_3\}.)

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You ask about the meaning of "isomorphic" in exercise 7.2:1.

An isomorphism  i: C --> D  of categories, like an isomorphism between
other mathematical objects, means a way of mapping the elements
comprising  C  bijectively to the elements comprising  D  so that the
structure is exactly preserved.  In detail, this means a bijection
i_{Ob}  of object-sets, and for each pair of objects  X, Y \in Ob(C),
a bijection  i_{X,Y}:  C(X,Y) --> D(i_{Ob}(X), i_{Ob}(Y)),  which
respects composition (and identity morphisms, though that follows
from the other conditions).

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In connection with Definition 7.4.4, you asked whether the concept
of an object such as a group  G  being U-small referred to the
whole structure, e.g., the group  (|G|, \mu_G, \iota_G, e_G),  being
a small set (a member of  U), or just the set  |G|.

Those two conditions are equivalent, since once  |G|  is U-small,
the map  \mu_G,  as a map  |G| x |G| --> |G|,  and hence a subset
of  (|G| x |G|) x |G|,  will also be U-small, and likewise for
\iota_G  and  e_G; hence the 4-tuple (|G|, \mu_G, \iota_G, e_G)
will be U-small.

Anyway, the intended meaning (in case there are cases
where the equivalence is evident) is that whole structure
(|G|, \mu_G, \iota_G, e_G)  is a U-small set.

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> ... you mention that in categories there is no real way to
> distinguish isomorphic objects. Is there any danger, then, of
> always looking at the skeleton category, which identifies
> isomorphic objects? ...

From an abstract point of view, no.  But in specific cases, it could
well be confusing.  For instance, suppose we are looking at the
category  Set,  and are considering the ordinals as objects of
that category.  Then the countably infinite ordinals (of which we
know there are uncountably many) all become isomorphic.  Suppose we
go to a skeleton category of  Set  in which the only representative
of that isomorphism class is  \omega.  We could still think about
the chain of inclusions of countable ordinals by taking a retraction
of Set to a skeleton, and looking at the images of the inclusion
maps  \alpha --> \beta  among the countably infinite ordinals as
certain endomorphisms of the object  \omega to which they have all
been retracted; and we could, for instance, describe the first
uncountable ordinal as the direct limit of that chain of endomorphisms
of \omega.  But it would be much easier to think of the category
Set  in which the countable ordinals are separate objects.

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You ask about (co)products that are preserved under all functors
(possibly assuming that the codomain category has coproducts).

Well, this will certainly be true of the coproduct of any 1-object
family!  My guess is that it will not be true in any other cases;
but I don't know.

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> As is mentioned in Section 8.1, all universal objects are initial
> objects in some category. But to show this, don't we need some
> abstract way of constructing a category in which a universal object
> is initial?  The examples given in 8.1 seem rather ad hoc, as they
> depend on the specific universal object being considered.

Well, we don't yet have a general definition of "universal object"!

In Chapter 4 we displayed a large bunch of interesting constructions
which seemed to have a feature of "universality" in common; and
in section 7.8 we found that some families of these could be gathered
under common category-theoretic descriptions.  In the first half of
Chapter 8 we will look at these more systematically.  From the point of
view of section 8.1, "universal object" might be defined as "structure
corresponding to an initial object in some category"; in later sections,
we look at specific sorts of properties that can be described as in
this way, but that have more natural descriptions in terms of other
categories, and how to pass between the two sorts of description.

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Regarding Definition 8.3.9 you asked,

> If we have three functors F,G,H, where G is a right adjoint of F
> and a left adjoint of H, is there anything interesting we can say
> about F and H? For instance, in the example you give with products,
> coproducts, and the diagonal functor, the product and coproduct
> functors are essentially dual to each other.

That the product and coproduct are "essentially dual" just means
that they are obtained by dual constructions -- left adjoint and
right adjoint -- from the diagonal functor, which has no left/right
asymmetry.  It refers to the way we look at them, not to properties
that they will have in a particular category.

But the question of whether, when a functor has a left and a right
adjoint, those two adjoints have properties relating them, is
interesting.  I don't know the answer, but I would guess that some
properties of  G,  concerning what distinctions it preserves and what
distinctions it loses, would force some dual properties on  F  and  H,  
which they would share.  But I don't know anything concrete about
this.

Alexey Zoubov has pointed out to me a result along these lines:
F is full and faithful if and only if H is full and faithful.
This is Lemma 1.3 in "Exponentiable morphisms, partial products
and pullback complements" by Roy Dyckhoff and Walter Tholen,
Journal of Pure and Applied Algebra, 49(1987) 103-116,
https://doi.org/10.1016/0022-4049(87)90124-1

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I meant to work the answer to your question into my lecture,
but didn't get to it.  You asked whether there is a notion 
of convergence associated with the direct limit.

I think that the main connection is conceptual -- that if  X
is the direct limit of a family of objects  X_i,  then the
successive  X_i  are "more and more like  X".  E.g., in the
case of a direct limit of groups, they in general have more
and more of the elements that will show up in  X,  and these
satisfy more and more of the relations that are satisfied there.

But it may be possible to turn this into topological convergence.
If we define a language in which there are symbols for all the
elements of  X,  and for the kinds of relations that these can
satisfy, then we might define a topology on any set of objects
of the indicated sorts, some elements of which are labeled with
some of the element-symbols, taking for a subbasis of open sets
the sets of objects characterized by having (or not having) an
element symbolized by each symbol, and satisfying (or not satisfying)
a given relation on such elements; and then I think the direct
limit  X  would be the topological limit of the system of "points"
X_i.

(In an abstract category, we might do something similar using
the existence of maps from various objects in the category
satisfying various composition relations; but I haven't thought
this through.)

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You ask about the second sentence after Theorem 8.8.9, saying
that if we assume that all functors from D to C have colimits,
then the isomorphism between the left and right expressions
in (8.8.10) becomes a case of Theorem 8.8.3.

I see that that does need a bit of explaining!

The first assertion of Theorem 8.8.3 says that if  F: C --> D
is a left adjoint functor, and  S: E --> C  has a colimit, then
lim_E FS = F(lim_E S).  (I will use "lim" in this e-mail for
the colimit symbol, "lim" with an arrow "-->" under it.)

In the application we want to make, the roles of  C  and  D  in
that theorem are taken by  C^D  and  C  respectively,  the role
of  F  by  lim_D,  and the role of  S  by  B(-,-),  regarded as
a functor  E --> C^D.  ("E" is the one symbol that translates to
itself!)  Then the formula  lim_E FS = F(lim_E S)  takes the form
lim_E lim_D B(-,-) = lim_D lim_E B(-,-).

(When I have time, after the end of the semester, I should add
a clarification of this to the online revised version of the
text.)

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> In proposition 8.9.3, it says that "In Set, direct limits commute
> with finite limits." This commutation should be understood as 'up to
> isomorphism', correct?

Well, limits and colimits are only defined up to isomorphism -- they
are objects with universal properties, and if one object has the
universal property, then anything isomorphic to it does.  It is true
that if we have specific constructions of the limit and colimit in one
order and in the other, then the "commutativity" statement means that
the two resulting objects are isomorphic (by an isomorphism that
respects the appropriate structure).

See Definition 8.8.1, last sentence and then first two sentences.

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Pointing to the contrast between Exercises 8.10:6 and 8.10:7, you
ask about general results on universal constructions in classes of
algebras with large sets of operations.

I don't know whether such results have been looked at.  In General
Algebra, it is most natural to consider algebras with small sets
of operations.  Complete lattices are most naturally thought of as
having meets and joins of arbitrary subsets; but operations on
subsets don't fit the techniques of General Algebra.  The best way
one can accommodate them is to treat them as "large" families of
operations on tuples of elements.  As such, they are more like the
kinds of structures studied in General Algebra than sets with
operations on subsets would be, but they still push the envelope.

I think "random" sorts of algebras with large classes of operations
will "almost never" have free objects; it is only certain sorts
that arise in special ways -- such as complete semilattices --
that have these.

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You asked about the fact, mentioned following Exercise 9.6:10,
that the concept of "heap", developed in that exercise, had
being rediscovered many times, and what could have led to that
repeated rediscovery.

I think that two contrasting aspects of the concept of heap are
relevant:  That it is fairly natural and moderately useful, but that it
does not invite study for its own sake, since the isomorphism classes
of heaps (other than the empty heap) are completely determined by
the isomorphism classes of the corresponding groups.  Naturalness and
usefulness leads people to discover the concept, but since the theory
completely reduces to that of groups, not many works get written about
them, and no one ends up specializing in them; so few people hear about
the concept, and if they need them, they often end up reinventing them.

When I rediscovered them I called them "isogroups"; see
p.60 of https://link.springer.com/article/10.1007%2FBF02188011 ,
line after display (5).  After that appeared, someone told me that
they had already been defined.

The same situation -- being natural and useful, but having a
theory that essentially reduces to other theories -- is also true
of preorders.  I don't know whether they too have been rediscovered
several times.  If not, perhaps the situations in which they come
up are sufficiently widespread that once a name was assigned to
the concept, enough people heard of it to prevent the "rediscovery"
syndrome.

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You write

> In the middle of section 9.7, it is said there is a connection
> between Lie algebras and Lie groups. Does this connection allow us
> to identify an algebra with a (possibly) group or vice-versa?

As the sentence preceding Exercise 9.7:6 suggests, it is a many-one
relationship:  The many different Lie groups that look alike in a
neighborhood of the identity have the same Lie algebra.  For each
finite-dimensional Lie algebra  L  over the real numbers, there is a
unique *simply connected* Lie group  G  having  L  as its Lie algebra.
Other Lie groups having  L  as their Lie algebra can be obtained
by dividing  G  by a discrete subgroup, and/or throwing in
more connected components.  For a trivial case:  if  L  is the
1-dimensional Lie algebra, which by (9.7.3) has zero bracket operation,
then  G  is the real line  R, while other Lie groups with the same
Lie algebra include the circle group  R/Z,  and various groups with
R  or  R/Z  as the connected component of the identity element.

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You ask why the assumption that  C  is a variety of algebras is
needed for condition (iii) of Definition 10.3.5 to be equivalent
to the other two.

If  C  is an arbitrary category with appropriate direct products
as in the first sentence of the definition, then "elements of A"
and "relations" satisfied by such elements aren't meaningful.

One might, of course, ask whether some weaker assumptions can
be made which would make those concepts meaningful.  One can
certainly do that.  If the conditions are too weak, the concepts
might be meaningful but the the equivalence could fail.  For
instance, if  C  is the category of all finite groups, one
can still speak of elements and relations, but a given system
of elements and relations might not determine a finite group
so such  X  and  Y  might not determine a representing object.
(E.g., one element and no relations give in the variety of all
groups the infinite cyclic group, but don't determine any object
of the category of finite groups.)  There are, however, conditions
weaker than that  C  be a variety of algebras, but strong enough
to make the equivalence hold.  But to develop these would require
that we introduce further concepts, and for simplicity, I have
restricted the main focus of the text to varieties of algebras.

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You ask about "a nice description" of the left adjoints of the
representable functors Monoid --> Monoid (section 10.6).

Well, they all have the nice formal description as functors
associating to each monoid  M  a monoid gotten by attaching
together a bunch of copies of the representing monoid  R, indexed
by the elements of  M,  with relations determined by the relations
of  M  and the comultiplication of  R.  But what the result looks
like for particular comonoids  R  can be complicated.

For the simplest interesting case:  The left adjoint of the functor
associating to a monoid its group of invertible elements is the functor
associating to a monoid  M  the result of adjoining an inverse to
every element.

The next-simplest interesting case is the left adjoint of the functor
associating to  M  the monoid of pairs  (a,b)  in  M  with  ab=e.
This adjoint adjoins to  M,  for every  a\in M,  an element a'  
such that  aa'=1,  in such a way that the map  a |-> a'  reverses
the order of multiplication.  Note that if in  M  one has  ab=e,
then in the new monoid,  b  will have both the left inverse  a
and the right inverse  b',  so it becomes invertible, so its left
inverse  a  also becomes invertible.  So the elements that were 1-sided
invertible  M  all become invertible in the new monoid; hence the
submonoid generated by those previously-1-sided-invertible elements
is embedded in a group.  Elements that were not previously
1-sided invertible need not become invertible, though they do
become 1-sided invertible.

I haven't studied those functors systematically ... .

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You ask about the meaning of "subobject" in the paragraph following
display (10.7.1).

Good point.  I guess I was implicitly relying on the fact that in
most real-world mathematical contexts, an idempotent endomorphism
of an object is a retraction to a subobject.  This isn't a formal
statement true or even meaningful in an arbitrary category; so that
comment should be considered a heuristic observation, suggesting what
we should look for.  It is valid with object taken to mean "category",
so it leads us to the right conclusion in this case.

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> Are there any examples of functors T: C^\op \to C such that T and
> TT have left and right adjoints but TTT does not?

I don't know.

One conceivable way to approach the question would be to have
C  be a direct product of, say, 4 categories C_0 x C_1 x C_2 x C_3,
and have  T  built out of functors  C_i --> C_{i+1}  (i=0,1,2),
such that some sort of bad behavior is reached only when we
carry C_0 into C_3.  I don't have anything detailed in mind, but
you might be able to go somewhere with the idea.

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