The term ‘syntax’ refers not to small categories such as algebraic theories or rings, but rather to their PRESENTATION by signatures or by polynomial generators, et cetera. The process of presentation is an adjoint pair quite distinct from the semantical adjoint pair: both adjoint pairs have a category of theories or of rings in common but are otherwise quite independent.
In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated).
Both of the functors
? ———————-> theories ————————–> Large categories
Syntax Semantics
are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera.
Happy New Year!
Let’s count the parts of a set map i.e. a function
f: A –> B
where domain set A = {cat, dog}, codomain set B = {flower, pet}, and with
f(cat) = pet
f(dog) = pet
Why bother counting parts of what looks like nothing more than an all too obvious labeling? Parts, where to begin, are profound. Take, for example, a singleton set
1 = {•}
which has two parts
0: 0 –> 1
1: 1 –> 1
each one of which is a 1-1 function. Just in case you are concerned about the fact that a function is required to assign to each element in the domain set an element in the codomain set and how on earth can there be a function
0: 0 –> 1
when there is no element in the empty domain set 0? Yes, that’s exactly the reason there is exactly one function from an empty set to a singleton set and that’s exactly the reason we also say there is exactly one function from an empty set to an empty set (of course there’s no arrow going from the empty domain set; not too long ago these two 1-1 functions with empty set as domain used to give me pounding headache; now I don’t feel ;) Of course, there is no function from a singleton set to an empty set because there is no element in the empty codomain set to which a function, to earn its existence, can assign the element in the domain singleton set (Conceptual Mathematics, page 30).
Now what about the parts of the function f: A –> B with which started? Parts are parts of an object. So we need to build a ‘universe of discourse’ with functions such as f: A –> B as objects.
Welcome to the Category of Set Maps (Conceptual Mathematics, pp. 144 – 5)! Around here objects are functions
f: A –> B
Maps from an object f: A –> B to an object g: C –> D are pairs of functions
<x, y>: f –> g
or, with little more precision, are commutative squares such as
B –y–> D
^………^
f |………| g
A –x–> C
satisfying
y f = g x
Here are some more cast-and-characters of our category of maps.
Returning to the question that got us started: counting the parts of an object (in the category of set maps)
f: A –> B
The parts of f: A –> B, somewhat analogous to the case of category of sets where parts of a set are 1-1 functions with the set as codomain, are monomorphisms (Conceptual Mathematics, pp. 339 – 42) in the category of set maps with
f: A –> B
as codomain object. So I have to count the number of monomorphisms with f: A –> B as codomain. Thanks to an exercise (Sets for Mathematics, page 116) which tells me what to look for—when looking for parts—I know that a part of
f: A –> B
is a pair of 1-1 functions
x: X –> A
y: Y –> B
satisfying
y g = f x
with g: X –> Y.
How many 1-1 functions are there with A = {cat, dog} as codomain set?
x1: 0 –> A, 0 = {}
x2: X –> A, X = {cat}
x3: X’ –> A, X’ = {dog}
x4: A –> A (identity function)
Along similar lines, there are also four 1-1 functions with B = {flower, pet} as codomain set i.e.
y1: 0 –> B, 0 = {}
y2: Y –> B, Y = {flower}
y3: Y’ –> B, Y’ = {pet}
y4: B –> B (identity function)
So we have 16 pairs of 1-1 functions with f: A –> B. Of these 16 pairs
<x1, y1>
<x2, y1>
<x3, y1>
<x4, y1>
.
.
.
<x4, y4>
how many satisfy
f xi = yj gij
with gij: xi –> yj and i, j = 1, 2, 3, 4.
Then there is the question of ‘how many gij are there for each i, j?’
Oh well, this’s what happens when an unstoppable force meets an immovable object!
Case I. i = 1, j = 1
x1: 0 –> A, 0 = {}
y1: 0 –> B, 0 = {}
g11: 0 –> 0
f: A –> B
Since f x1 = y1 g11 (think multiplying with zero as in n × 0 = 0 × 0)
K, we have a part <x1, y1> of f: A –> B.
Case II. i = 2, j = 1
x2: X –> A, X = {cat} and x2(cat) = cat
y1: 0 –> B, 0 = {}
Unfortunately there’s no function g21 from a singleton set X to an empty set 0.
So <x2, y1> is not a part of f: A –> B.
Case III. i = 1, j = 2
x1: 0 –> A, 0 = {}
y2: Y –> B, Y = {flower} and y2(flower) = flower
g12: 0 –> Y
f: A –> B
Since f x1 = y2 g12 (think multiplying with zero as in n × 0 = m × 0)
OK then we got another part <x1, y2> of f: A –> B. This g12: 0 –> Y, thinking of the domain and codomain as ‘present’ and ‘past’ respectively, is how you talk about people who didn’t make it to the present (cf. Two-Stage Variable Sets; Sets for Mathematics, pp. 114 – 9).
Lucky you—I, unlike that bunny that keeps going and going, sleep like 12hrs/day; I kinda see adjoint functors high up on the cloud atlas (Exercise 6.14; Sets for Mathematics, page 119) from where we are at (Exercise 6. 6; Sets for Mathematics, page 116) if I squint just right 
Being the sadist that I’m I’ll be back to go through all the cases and correct the mistakes I made here.
Happy hump day!
Dear Posina,
[…]
Truth and the terminal object:
Concerning your question last month about the role of the terminal object in calculating the truth value object of a topos: for example, you note correctly a similarity between the topos of abstract sets and the topos whose objects are maps of abstract sets; this similarity does not extend to any kind of directed graphs, nor to group actions etc. (as you noted). I think you have re-discovered the important class of toposes known as ‘localic’. Recall that an important method of analyzing the inside of objects is the employment of a small subcategory of figure shapes. The basic requirement on this subcategory in its relation to the whole is that for any two maps f, g: X —> Y, one can conclude that f = g provided fx = gx for every figure x: A —>X with A in the little subcategory. (The actual structure of the inside of X is revealed by incidence relations a: A’ —> A which relate particular figures by x’ = xa and ramifications thereof.) For example, in categories of graphs, typically a two-object subcategory suffices. In general, the truth value object is determined by knowing all the subobjects of all these preferred figure shapes and how these substitute along incidence relations. Of course, ‘points’ in the narrow sense are just 1-shaped figures, so points of the truth value object correspond to subobjects of 1. In case the subobjects of 1 are sufficient to distinguish maps in the above sense, the whole topos is called ‘localic’, and this subcategory can be visualized as consisting of various open regions in a topological space. At the level of the 2-category of set-based toposes, your one example (which has only one region besides the two trivial ones) has a certain universality among all the localic toposes. Contrary to a wide-spread misconception, the localic toposes are not really typical, but do incorporate one important notion of ‘variable set’. On the other hand, the other large class of cohesive toposes is essentially disjoint from the localic ones, even though there are many relationships between the two classes.
I hope that the above account is helpful.
————————-
Dear Professor F. William Lawvere,
The relation between terminal objects and truth value objects varies across categories. For example, in the category of sets one could [exhaustively] characterize the truth value object if we are given the terminal object (cf. the 1-1 correspondence between parts of the terminal object and points of the truth value object), but it’s not the case in the category of graphs. On the other hand, the relation between truth value object and terminal object in the category of two-stage variable sets seems to be like that in the category of sets.
I am wondering if there’s something about a category that speaks directly to the way truth value object corresponds to the terminal object in the category (please forgive me for this diffuse question). Naively speaking, I am thinking that studying this correspondence can help in the calculation of the truth value object from the terminal object in various categories.
I’d appreciate very much your clarifications and guidance in my study of truth.
My heart-felt congratulations to you on the occasion of being named Fellow of the American Mathematical Society!
Happy Ugadi! (Happy New Year’s Day in my state of Andhra Pradesh)
Thanking you,
Yours sincerely,
Dear Posina,
There are a couple of cautionary remarks I would like to make:
For one thing, Conceptual Mathematics is not as such a field of study, but only a hopefully helpful guide for the beginner to initiate the study of space and quantity and the human struggle to understand their relationship, with the Science explicitly isolated and developed by Eilenberg, Mac Lane, Grothendieck, Kan, Isbell, Schanuel, et al . You recognized this by your inclusion of more of my writings, although a fairer representation would certainly cite the writings of others too, and even some actual applications of Category Theory. The title [of the textbook] actually arose because there was an NSF proposal for another book on ‘Discrete Mathematics’, which had been for some years a method of promoting one-sidedness in mathematics education. Steve Schanuel and I said ‘No, we want to teach the basics of methods that apply to both discrete and continuous mathematics and to their relationships’. We chose the word ‘Conceptual’ to express this, although some friends wisely warned us that another common mainstream misuse of ‘conceptual’ in School curricula is to suggest imprecise mysticism as opposed to concrete calculations. Thus, the second thing I want to post as a warning is that there is a trend (fortunately still a minor trend) to use our book, not as science, but as a source of subjective highs. […]
A third point has to do with the dialectics of the application of the category of abstract sets to real situations, for ex. choosing which breakfasts to serve to one’s friends. In connection with that and similar examples in the text, we had some misgivings. We tried to emphasize that there are three steps, the extra-mathematical assignment of names of real things to points in definite sets, the application of composition, products, etc. to those definite sets, and then the reading off of real conclusions. For example, reading off the fibers of a map will tell us things like how many friends received the same kind of breakfast. But since in the extreme case, all singleton sets are uniquely isomorphic, there is no value or information to be derived from displaying an arrow from one labeled singleton set to another, so it is non-mathematical to promote that illusion. Science will take us back to reality.
Thank you again,
best wishes
I hope you enjoy walking all by yourself at 3 in the morning–it’s going to be a long walk, along two-stage variable sets, to adjoint functors! Hopefully we will get to ask how they got here–did they sin as much as we did to be amongst us?
Sin, for some inexplicable reason, reminds me of conservatives. As we all know, for each conservative there is a liberal that the conservative was during the previous stage [of being sensible]. So a set of people undergoing the transition from the previous liberal stage of having fun to the present conservative stage of panicking about the possibility of somebody somewhere having fun can be analysed in terms of two sets Conservatives, Liberals and a function
f: Conservatives –> Liberals
specifying for each element x in Conservatives the element in Liberals that x was during the previous liberal stage. We may note here that not all liberals graduate into conservatives (some stay young forever; Sets for Mathematics, pp. 114 – 9).
Given that the objects of the category of two-stage variable sets (Sets for Mathematics, pp. 114 – 9) are functions
ξX: XU <– X1
is it correct to think, in view of the fact that the functions ξX need not necessarily be onto or 1-1, that the [underlying] connection from a previous stage to the present stage
U –> 1
can not be a function.
If, for example, I take
ξX: XU <– X1
with X1 = {mother, child}, XU = {pregnant woman} and
ξX(mother) = pregnant woman
ξX(child) = pregnant woman
then it is not clear how one could construe the physical process from the past (with one pregnant woman) to the present (of a mother and a child) as a function. Having used ‘life’ to illustrate the above case of onto ξX, we now use ‘death’ to illustrate the case of 1-1 ξX i.e.
ξX: XU <– X1
with X1 = {Posina}, XU = {Subrahmanyam, Rayudu}. Since my brother is no more, there is only one element ‘Posina’ in the present with
ξX(Posina) = Rayudu
reminding ‘Posina’ (in X1) of the ‘Rayudu’ (in XU) that he was sometime in the long-gone past. Once again it is not clear how the process from the past (of two brothers) to the present (with just one left) can be thought of as a function. All this detour into life-death is simply to say that it seems like there’s a difference between the category of two-stage variable sets (Sets for Mathematics, pp. 114 – 9) and the category of maps (Conceptual Mathematics, pp. 144 – 5), and I have no idea what exactly the difference is!
Setting aside the sadness, here’s an happy study with
X1 = {dog, cat}
XU = {kitten, puppy}
and the function
ξX: XU <– X1
specifying for each element x of the domain set X1 the element that x was during its playful years (in the codomain set XU) before it got leashed into adulthood i.e.
ξX(cat) = kitten
ξX(dog) = puppy
Now when I look at this object
ξX: XU <– X1
what I see depends on what I attend to: attending to the ‘dog’ (along with the ‘puppy’ that it was) looks like a map with the object ξX as codomain object of the map
11 –> ξX
and if we focus-in a little bit more, then it looks like a commutative square
1 –puppy–> XU
^ ^
11 | | ξX
1 –dog–> X1
satisfying
puppy 11 = ξX dog
Here there’s a parallel to the category of sets. Elements of a set, say,
X = {x1, x2}
are in 1-1 correspondence with the functions
x1: 1 –> X
x2: 1 –> X
from the terminal object (a singleton set 1 = {•}) of the category of sets. Similarly elements of an object X of the category of two-stage variable sets are maps to X from the terminal object
11: 1 –> 1
Just to be clear, an object in the category of sets is a set. (Why are categories named after their objects? It has, my porous memory tells me, something to do with NATURAL, and that’s all I know.) An object in the category of two-stage variable sets is a function
ξX: XU <– X1
from a domain set X1 to a codomain set XU (see also the category of maps; Conceptual Mathematics, pp. 144 – 5). Terminal object of the category of two-stage variable sets is the function
11: 1 –> 1
from 1 (= {•}) to 1. In other words, there is exactly one map
f: 1 <– X
from every object
ξX: XU <– X1
in the category of two-stage variable sets to
11: 1 <– 1
i.e. a unique commutative square
1 <–fU– XU
^ ^
11 | | ξX
1 <–f1– X1
(fU sending every element of XU to the only element of 1 and f1 sending every element of X1 to the only element of 1) satisfying
fU ξX = 11 f1
I almost forgot about composition.
Given a pair of composable maps
X –f–> Y and Y –g–> Z
i.e. two commutative squares
XU –fU–> YU
^ ^
ξX | | ξY
X1 –f1–> Y1
and
YU –gU–> ZU
^ ^
ξY | | ξZ
Y1 –g1–> Z1
satisfying
fU ξX = ξY f1
gU ξY = ξZ g1
We say that the composite map
X –f –> Y –g –> Z = X –gf –> Z
is the composite square
XU –fU–> YU –gU–> ZU
^ ^ ^
ξX | | ξY | ξZ
X1 –f1–> Y1 –g1–> Z1
i.e. the commutative square
XU –gUfU–> ZU
^ ^
ξX | | ξZ
X1 –g1f1–> Z1
satisfying
gU fU ξX = ξZ g1 f1
But does it? Let’s see—we are given
fU ξX = ξY f1 and gU ξY = ξZ g1
and we have to see if
gU fU ξX = ξZ g1 f1
Starting with gU fU ξX which looks like
XU –fU–> YU –gU–> ZU
^
| ξX
X1
which equals gU ξY f1 which in turn looks like
YU –gU–> ZU
^
| ξY
X1 –f1–> Y1
which in turn equals ξZ g1 f1 which in turn looks like
ZU
^
ξZ |
X1 –f1–> Y1 –g1–> Z1
All of the above can be summed up in words as follows:
going from X1 to ZU via the (up-right) path of gU after fU after ξX
is same as
going from X1 to ZU via the (right-up) path of ξZ after g1 after f1
I don’t know what you are thinking, but I’m like, after what feels like eons, I finally did one exercise (Exercise 6.4; Sets for Mathematics, page 115); I know it’s not even an atom in the grand scheme of things or is it Next exercise to flex my muscle is Exercise 14 (Conceptual Mathematics, page 144), which, for really good reasons, I love!
(Slowly but surely we’ll get to adjoint functors one exercise after one exercise in the category of two-stage variable sets! Sorry about the clumsy diagrams above; it looks like I’m better off going back to attaching word.docs)
I’m all for, lest you conclude otherwise, Against Interpretation!
Given a couple of sentences made up of few words such as
Posina is in India.
India is in China.
I can translate from the given English into Swedish as
Posina är i Indien.
Indien är i Kina.
Translation, moving away from the given particulars, involves
FW: WordsE –> WordsS
mapping words in English into words in Swedish such as
Posina |-> Posina
is |-> är
and
FS: SentencesE –> SentencesS
mapping sentences as in
Indien är i Kina.
^
|
India is in China.
In mapping sentences we make sure that subjects go to subjects and objects go to objects; otherwise we might end up translating
India is in China.
as
Kina är i Indien.
and find that China is not too happy with our translation, which is rather serious—so let’s formalize it for all future generations to learn before it’s too late!
I think of a sentence as an arrow
• –> •
with the dot on the left as subject and the dot on the right as object of the sentence i.e.
subject –Sentence–> object
Translation FS of one sentence
A –p–> B
into another
C –q–> D
should respect the subject / object structure of sentences i.e.
if FS(p) = q
then [it better be] FW(Sj(p)) = Sj’(FS(p))
which says something like:
the subject (Sj) of sentence p is mapped (by FW) to the subject (Sj’) of the sentence (q) to which the sentence p is mapped by FS.
So is the case with objects i.e.
FW(Oj(p)) = Oj’(FS(p))
Just in case you don’t like equations, here it is in pictures
B –FW–> D
^ ^
| |
p =FS=> q
| |
A –FW–> C
In little bit higher resolution it looks like the commutative square below in the case of subjects
WordsEng –FW–> WordsSwe
^ ^
SjE | | SjS
| |
SentencesEng –FS–> SentencesSwe
and in the case of objects
WordsEng –FW–> WordsSwe
^ ^
OjE | | OjS
| |
SentencesEng –FS–> SentencesSwe
If you are now like—I miss equations
SjS(FS) = FW(SjE)
OjS(FS) = FW(OjE)
or better yet, in plain words, subject / object of a translated sentence is same as the translated subject / object of the sentence translated.
Given
Posina is in India. India is in China.
I’m tempted to think
Posina is in China.
This thought expressed in arrow-language looks like two arrows
Posina –is in–> India. India –is in–> China.
composed into one arrow
Posina –is in–> China.
Having had enough of lost-in-translation, we make sure translation (F) preserves composition of sentences i.e.
F(qp) = F(q) F(p)
which says something like:
composite of translated sentences is same as the translation of the composite of sentences
Now all this TRANSLATION is looking lil like FUNCTOR (Conceptual Mathematics, page 167), which is an interpretation of one category into another.
CATEGORY, when one looks at the definition (Conceptual Mathematics, page 21), has two things:
OBJECTS
MAPS
Functor F takes objects to objects and maps to maps i.e.
FOb: Objects –> Objects
FMp: Maps –> Maps
Objects and maps in a category are related to one another via structural maps:
domain: Maps –> Objects
codomain: Maps –> Objects
identity: Objects –> Maps
composition: Maps3 –> Maps
and functors from one category to another respect this structure of categories i.e.
FOb(dom) = dom’(FMp)
FOb(cod) = cod’(FMp)
FMp(identity) = identity’(FOb)
FMp(gf) = FMp(g) FMp(p)
If you translate from English to Swedish, then I see no reason not to translate from Swedish to English; and in translating from English to Swedish and from Swedish to English i.e.
F: English –> Swedish
G: English <– Swedish
it’s possible to end with what we started with:
India is in China.
^
|
Indien är i Kina.
^
|
India is in China.
which is boring enough to warrant careful examination of pairs of opposed functors
F: A –> B
G: A <– B
There are, among these opposed functors, some special pairs called ADJOINT FUNCTORS, which deserve our special attention. I though of getting a feel for adjoint functors in terms of
and this
there is at most one inclusion between two parts
there is at most one coarsening between two partitions
is how close I got to it 
One thing I don’t like about story-telling is them stories sitting still every time the teller goes on a smoke break; wouldn’t it be nice if the story can script itself without waiting for me to exhale?
Imagine, upon reading the definition of truth value object (Conceptual Mathematics, page 337), finding that the textbook ends then and there—no more spilled-ink to make sense of. There’s, within this doomsday scenario, enough momentum built, definition-after-definition beginning (where shall we) with terminal objects such as 1 (Conceptual Mathematics, pp. 225 – 9), into the story to keep characters moving in the heads heading home long after the curtain dropped on
true: 1 –> Ω
What if I reverse an arrow
–>
into the arrow
<–
How about an arrow from Ω
X <– Ω
instead of the arrow to Ω
X –> Ω
Reversing an arrow
–>
into the arrow
<–
doesn’t cost whole lot of consciousness and that’s all(?) it takes to get to the definition of 0 (empty set):
there is, for each set X, one function 0 –> X
from the definition of 1 (singleton set):
there is, for each set X, one function 1 <– X
Just as I was about to write-off 1 as bland (after all there is just one function to 1; Conceptual Mathematics, page 302), points i.e. functions from 1
1 –> X
are all over the place spicing up the story (Conceptual Mathematics, page 19, pp. 230 – 5).
With this preamble in place, let’s indulge in some head-scratching: the definition of truth value object is beaming with arrows (waiting to be reversed?) such as
(i) a given part true: T –> Ω
(ii) every part g: U –> X
(iii) exactly one arrow f: X –> Ω
or, shall we say, loaded with heavyweights such as
Beginning with belongs to, what do we get if we reverse arrows in
x belongs to y
I’d have never guessed
determined by
to be the opposite of
Then again it’s not like I suspected the initial object
0
to be the opposite of the terminal object
1
or SUM to be the opposite of PRODUCT (Conceptual Mathematics, page 254, 284). I have nothing to show even after looking in every gyri and sulci of my big male-brain for the opposite of PART (Sets for Mathematics, pp. 37 – 8).
That’s all the self-indulgent blather my shrink can stand this Sunday
We say
f: A –> Y
is determined by
g: A –> X
whenever
f = pg (Conceptual Mathematics, pp. 45 – 9, 68 – 80, 370 – 1).
Note to self: One of these days you need to stop talking words or at least start drawing diagrams! Until then I need not highlight the fact that science is preoccupied in every step of its long-winding road to reality with DETERMINATION. Watching bodies-in-motion one wonders about distances traveled by the moving bodies
f: Moving bodies –> Distances
and thinks of noting down the duration of motion
g: Moving bodies –> Durations
and hopefully finds that there is a function
h: Durations –> Distances
(assigning to each duration of motion the distance traveled by moving bodies in that duration) such that
f = hg
in which case we say
f is determined by g
Posina Venkata Rayudu
