##### The unreasonable power of the lifting property in elementary mathematics
##### misha gavrilovich
in memoriam: evgenii shurygin
epigraph:
In the following sections we bring up instances of human and animal behavior
which are, on one hand, miraculously complicated, on the other hand they have
little, if any, pragmatic (survival/reproduction) value. From this we conclude
that since the corresponding features of ergobrains were not the primarily targets
specifically selected for by the evolution, they are due to internal constrains on
possible architectures of unknown to us functional "mental structures".
Gromov, Ergobrain
#### abstract:
We illustrate the generative power of the lifting property as a means of
defining natural elementary mathematical concepts by giving a number of examples in
various categories, in particular showing that many standard elementary notions
of abstract topology can be defined by applying the lifting property to simple
morphisms of finite topological spaces. We include some speculations on the
wider significance of this.
# Introduction.
The purpose of this short note is to draw attention to the following
observation which we find rather curious:
a number of elementary properties from a first-year course
can be defined category-theoretically by
repeated application of a standard category theory trick,
the Quillen lifting property,
starting from a class of explicitly given morphisms,
often consisting of a single (counter)example
In particular, several elementary notions of topology
have Kolmogorov complexity of several bytes
in a natural category-theoretic formalism,
e.g. compactness is "({0}-->{0->1})^r_{<5}^lr",
connectedness is "({0,1}-->{0=1})^l",
dense image is "({1}-->{0->1})^l".
We suggest it appears worthwhile to try to develop a formalism,
or rather a very short program (kilobytes) based on such a formalism,
which supports reasoning in elementary topology.
These observations arose in an attempt to understand ideas of
Misha Gromov [Memorandum Ergo] about ergologic/ergostructure/ergosystems.
Oversimplifying, ergologic is a kind of reasoning which helps to
understand how to
generate proper concepts, ask interesting questions, and, more generally,
produce interesting rather than useful or correct behaviour.
He conjectures there is a related class of
mathematical, essentially combinatorial, structures,
called _ergostructures_ or _ergosystems_,
and that this concept might eventually
help to understand complex biological behaviour including learning
and create mathematically interesting models of these processes.
We hope our observations may eventually help to uncover
an essentially combinatorial reasoning behind elementary topology,
and thereby suggest an example of an ergostructure.
Structure of the paper. A mathematically inclined reader might want to read
only the first two sections with miscellaneous examples of lifting properties
and a combinatorial notation for elementary properties of topological spaces.
A logician might want to read in the third section our suggestions towards
a theorem prover/proof system for elementary topology based on diagram
chasing. Appendix A states separation axioms in terms of lifting properties
and finite topological spaces.
Finally, the last sections attempts to explain our motivation and says
a few words about the concept of ergostructure by Misha Gromov.
# The lifting property: the key observation
For a property C of arrows (morphisms) in a category, define
C^l := { f : for each g in C f /_ g }
C^r := { g : for each f in C f /_ g }
C^lr:=(C^l)^r, ...
here f /_ g reads " f has the left lifting property wrt g ",
" f is (left) orthogonal to g ",
i.e. for f:A-->B, g:X-->Y,
f /_g iff for each i:A-->X, j:B-->Y such that ig=fj ("the square commutes"),
there is j':B-->X such that fj'=i and j'g=j ("there is a diagonal
making the diagram commute").
The following observation is enough to reconstruct all the examples in this
paper, with a bit of search and computation.
Observation.
A number of elementary properties can be obtained by repeatedly passing
to the left or right orthogonal C^l, C^r, C^lr, C^ll, C^rl, C^rr,...
starting from a simple class of morphisms, often
a single (counter)example to the property you define.
A useful intuition is to think that the property of left-lifting against a
class C is a kind of negation of the property of being in C, and that
right-lifting is another kind of negation. Hence the classes obtained from C
by taking orthogonals an odd number of times, such as C^l, C^r, C^lrl, C^lll
etc., represent various kinds of negation of C, so C^l, C^r, C^lrl, C^lll each
consists of morphisms which are far from having property C.
Taking the orthogonal of a class C is a simple way to define a class of
morphisms excluding non-isomorphisms from C, in a way which is useful in a
diagram chasing computation.
For example, the notion of isomorphism can be obtained starting from the class
of all morphisms, or any single example of an isomorphism:
(Isomorphisms) = (all_morphisms)^l = (all_morphisms)^r = (h)^lr = (h)^rl
where h is an arbitrary isomorphism.
Example. Take C={ {}-->{o} } in Sets and Top.
A-->B /_ {}-->{o} iff A=B={} or A=/={}, B arbitrary. Indeed,
if A=/={}, there is no map i:A-->X={} and the lifting property
holds vacuously; if A={}=/=B, there exist unique maps i:A={}-->X={},
j:B-->Y={o}, but no map j':B-->X={} as B=/={} by assumption.
{}-->{o} /_ g iff g:X-->Y is surjective; indeed,
the map j:B={o}-->Y picks a point in Y and j':B={o}-->X
picks its preimage as j'g=j; the other condition fj'=i:{}-->X
holds trivially. Thus ({}-->{o})^r is the class of surjections.
In Sets, ({}-->{o})^rl is the class of injections, i.e.
f /_ g for each surjection g iff f is injective; indeed,
for such f and g the following is well-defined:
set j'(b)=i(f^{ -1}(b)) for b in Im f, and j'(b)=g^{ -1}(j(b))
otherwise; for injective f j'(b) does not depend on the choice
of a preimage of b, and for g surjective a preimage always exists.
In Top, ({}-->{0})^rl is the class of maps of form A-->A\/D, D is discrete;
given a map A-->B, consider A-->B /_ ImA\/(B\A)-->B where ImA\/D-->B
denotes the disjoint union of the image of A in B with induced topology,
and B\A equipped with the discrete topology.
In both Sets and Top, ({}-->{o})^rr is the class of subsets, i.e.
injective maps A(-->B where the topology on A is induced from B.
Toying with the observation leads to the examples in the claim below
which is trivial to verify, an exercise in deciphering the notation
in all cases but (vii) proper.
Claim 1.
(i) ({}-->{o})^r, (0-->R)^r, and {0-->Z}^r are the classes of surjections in
Sets, R-modules, and Groups, resp,
(where {o} is the one-element set, and in the category of groups, 0 denotes the trivial group)
(ii) ({a,b}-->{o})^l=({a,b}-->{o})^r, (R-->0)^r, {Z-->0}^r are the classes of
injections in Sets, R-modules, and Groups, resp,
(iii) in R-mod, a module P is projective iff 0-->P is in (0-->R)^rl
a module I is injective iff I-->0 is in (R-->0)^rr
(iv) in the category of groups,
a finite group H is nilpotent iff H-->HxH is in { 0-->G : G arbitrary }^lr .
a finite group H is solvable iff 0-->H is in { 0-->A : A abelian }^lr= { [G,G]-->G : G arbitrary }^lr
a finite group H is of order prime to p iff H-->0 is in {Z/pZ-->0}^r
a finite group H is a p-group iff H-->0 is in {Z/pZ-->0}^rr
a group H is torsion-free iff 0-->H is in { nZ-->Z: n>0 }^r
a subgroup A is pure in a group H iff A-->H is in { nZ-->Z : n>0 }^r
(v) in the category of metric spaces and uniformly continuous maps,
a metric space X is complete iff {1/n}_n-->{1/n}_n\/{0} /_ X-->{0}
where the metric on {1/n}_n and {1/n}_n\/{0} is induced from the
real line
a subset A (= X is closed iff {1/n}_n-->{1/n}_n\/{0} /_ A-->X
(vi) for a connected topological space X, each function on X is bounded
iff {}-->X /_ |_|_n (-n,n) --> R in the category of topological spaces
(vii) in the category of topological spaces (see notation defined below),
({1}-->{0->1})^l is the class of maps with dense image,
({1}-->{0->1})^lr is the class of closed inclusions A (= X, A a closed subset of X,
({}-->{o})^l is the class of maps A-->B such that A=B={} or A=/={}, B arbitrary,
({}-->{o})^ll is the class of maps A-->B such that either A={} or the map is an isomorphism
({}-->{0})^r is the class of surjections,
({}-->{0})^rl is the class of maps of form A-->A\/D, D is discrete
({}-->{o})^rr is the class of subsets, i.e. injective maps A(-->B where the topology on A is induced from B.
(({0}-->{0->1})^r_{<5})^lr is roughly the class of proper maps
(see below).
Proof. In (iv), we use that a finite group H is nilpotent iff the diagonal {
(h,h) : h in H } is subnormal in HxH. (vii) is discussed below. QED.
Problem 0. Find more examples of meaningful lifting properties in various categories.
Play with natural classes of morphisms to see whether their iterated orthogonals
are meaningful.
Remark. Claim (v) shows that a computer-generated proof in (Ganesalingam, Gowers; Problem 2)
of the claim that a closed subspace of a complete metric space is necessarily complete
translates to two applications of a diagram chasing rule
corresponding to the lifting property. We say more about this later.
Remark. Claim demonstrates that a number of elementary notions can be concisely
expressed in terms of a simple diagram chasing rule. However,
it appears there is no (well-known) logic or proof system
based on diagram chasing in a category. We make some suggestions towards
such a proof system in the last two sections.
# Elementary topological properties via finite topological spaces
First we must introduce notation for maps of finite topological spaces
we use. Two features are important for us:
1. it reminds one that a finite topological space is a category
(degenerate if you like)
2. it does not allow one to talk conveniently about non-identity
endomorphisms of finite topological spaces. We hope this may help define
a decidable fragment of elementary topology because there is
a decidable fragment of diagram chasing without endomorphisms, see [GLZ].
A topological space comes with a _specialisation preorder_ on its points: for
points x,y in X, x =< y iff y in cl x , or equivalently, a category whose
objects are points of X and there is a unique morphism x->y iff y in cl x.
For a finite topological space X, the specialisation preorder or equivalently
the category uniquely determines the space: a subset of X is closed iff it is
downward closed, or equivalently, there are no morphisms going outside the
subset.
The monotone maps (i.e. functors) are the continuous maps for this topology.
We denote a finite topological space by a list of the arrows (morphisms) in
the corresponding category; '<->' denotes an isomorphism and '=' denotes
the identity morphism. An arrow between two such lists denotes
a continuous map (a functor) which sends each point to the correspondingly
labelled point, but possibly turning some morphisms into identity morphisms,
thus gluing some points.
Thus
{a,b}-->{a->b}-->{a<->b}-->{a=b}
denotes
(discrete space on two points)-->(Sierpinski space)-->(antidiscrete space)-->(single point)
In {a->b}, the point a is open and point b is closed.
Let
C_{{0->1})^r_{<5})^lr is almost the class of proper maps, namely
a map of T4 spaces is in the class iff it is proper
({1}-->{0->1})^l is the class of maps with dense image,
({1}-->{0->1})^lr is the class of maps of closed inclusions A (= X, A is closed
({}-->{0})^r=({0}-->{0<->1})^l is the class of surjections,
({}-->{0})^rl is the class of maps of form A-->A\/D, D is discrete
({}-->{0})^rll = ({a}-->{a,b})^l is the class of maps A-->B such that
each connected component of B intersects Im A.
({}-->{o})^l is the class of maps A-->B such that A=B={} or A=/={}, B arbitrary,
({}-->{o})^ll is the class of maps A-->B such that either A={} or the map is an isomorphism
({}-->{o})^rr is the class of subsets, i.e. injective maps A(-->B where the topology on A is induced from B.
({a<->b}-->{a=b})^l is the class of injections,
({a->b}-->{a=b})^l is the class of maps f:X-->Y
such that the topology on X is induced from Y
({a,b}-->{a=b})^l describes being connected, and in fact
is the class of maps f:X-->Y
such that pi_0(X) (--> pi_0(Y) is injective
(at least when the connected components of
both X and Y are open and therefore pi_0(X), pi_0(Y)
are both nicely defined)
({a<->b}-->{a=b})^r fibres are T0 spaces
({a->b}-->{a=b})^r fibres are T1 spaces
({a,b}-->{a=b})^r is the class of injections
({a}-->{a<->b})^l is the class of surjections,
({a}-->{a,b})^l also describes being connected, and in fact
is the class of maps A-->B such that
each connected component of B intersects Im A
({a}-->{a<->b})^r is the class of surjections ,
({b}-->{a->b})^l something T1-related but not particularly nice
({a}-->{a->b})^l something T0-related
({a}-->{a,b})^l is the class of maps f:X-->Y
such that either X is empty or f is surjective
Proof. All items are trivial to verify, with the possible exception of the first
item. [Bourbaki, General Topology, I\S10.2, Thm.1(d), p.101] gives a characterisation
of proper maps by a lifting property with respect to maps associated to
ultrafilters. Using this it is easy to check that each map in ({0}-->{0->1})^r_{<5}
being closed, hence proper, implies that each map in (({0}-->{0->1})^r_{<5})^lr is proper.
A theorem of [Taimanov] states that for a compact Hausdorff space K,
the map K-->{*} is in (({0}-->{0->1})^r_{<5})^lr, and its proof generalises to
give that a proper map between normal Hausdorff (T4) spaces is in the class.
QED.
Conjecture. In the category of topological spaces,
(({0}-->{0->1})^r_{<5})^lr is the class of proper maps.
Remark. It is easy to see that (({0}-->{0->1})^r_{{0->1})^r_{m>3 such that the inclusion is strict.
An example using cofinite topology (suggested by Sergei Kryzhevich) suggests
that (({0}-->{0->1})^r_{<4})^lr (=/= (({0}-->{0->1})^r_{<5})^lr .
Problem. Calculate (({1}-->{0->1})^r_{<5})^lr, (({1}-->{0->1})^lrr, and
({a<-U->x<-V->b}-->{a<-U=x=V->b})^lr .
Could either be viewed as a "definition" of the real line?
Remark. Urysohn lemma and Tietze extension theorem explains why we hope it might have something to
do with the real line, as follows.
Note a map f of finite spaces is open iff f is in ({1}-->{0->1})^r,
and that {a<-U->x<-V->b}-->{a<-U=x=V->b} is an open map.
A space X is normal (T4) iff {}-->X /_ {a<-U->x<-V->b}-->{a<-U=x=V->b}, hence
the Uryhson lemma can be stated as follows:
{}-->X /_ {a<-U->x<-V->b}-->{a<-U=x=V->b} iff {}-->X /_ {0'} \/ [0,1] \/ {1'} ---> {0=0'->x<-1=1'}
where points 0',0 and 1,1' are topologically indistinguishable in
{0'} \/ [0,1] \/ {1'},
the closed interval [0,1] goes to x, and 0,0' map to point 0=0', and 1,1'
map to point 1=1'.
Tietze extension theorem states that for a normal space X and a closed subset A of X,
A-->X /_ R-->{0} , i.e. in notation
R-->{0} is in ( ({1}-->{0->1})^lr /\ { A-->X : {}-->X /_ {a<-U->x<-V->b}-->{a<-U=x=V->b} )^r
Note that {}-->X /_ [0,1] ---> {0->x<-1} is the stronger property X is
perfectly normal. A normal space X is perfectly normal provided
each closed subset of X is the intersection of a countably many open subsets.
For example, is {0'} \/ [0,1] \/ {1'} ---> {0=0'->x<-1=1'} in
({a<-U->x<-V->b}-->{a<-U=x=V->b}^lr (= (({1}-->{0->1})^r_{<5})^lr ?
Problem. Is there a model category or a factorisation system of interest
associated with any of these lifting properties, for example compactness/properness?
Many of the separation axioms can be expressed as lifting properties with respect to maps
involving up to 4 points and the real line, see [Appendix A].
## Elementary topology as diagram chasing computations with finite categories
The Hausdorff axioms of a topological space can be seen as
diagram chasing rules for manipulating diagrams
involving notation such as
{x}-->X, X-->{x->y}, X-->{x<->y},
in the following straightforward way; cf. [Gavrilovich, Elementary Topology,\S.2.1] for more
details.
As is standard in category theory, identify a point x of a topological space X
with the arrow {x}-->X, a subset Z of X with the arrow X-->{z<->z'},
and an open subset U of X with the arrow X-->{u->u'}.
With these identifications, the Hausdorff axioms of a topological space become
rules for manipulating such arrows, as follows.
_Both the empty set and the whole of X are open_ says that the compositions
X-->{c}-->{o->c} and X-->{o}-->{o->c}
behave as expected (the preimage of {o} is empty under the first map,
and is the whole of X under the second map).
_The intersection of two open subsets is open_ means the arrow
X--->{o->c}x{o'->c'}
behaves as expected (the intersection is the preimage of (o,o') in
{o->c}x{o'->c'} ).
Finally, _a subset U of X is open iff each point u of U has an open
neighbourhood inside of U _
corresponds to the following diagram chasing rule:
for each arrow h:X--->{U<->U'},
{U->U'}
/\ |
(exist) / |
/ |
/ |
/ v
X --h-->{U<->U'}
iff for each arrow {u}--->X,
{u}------>{u->U<->U'}
| /\ |
|(exist)/ |
| / |
| / |
v / v
X --h--->{u=U<->U'}
here the arrows {u}----->{u->U<->U'}
and {U->U'}--->{U<->U'} denote the obvious maps.
%\begin{itemize}
%\item[($3_\rightarrow$)] %a set $U$ is open iff for every point $u\in U$ there
%is an open subset $U'$ such that $u\in U'\subset U$.
%for each arrow $X\xra[\xi_U]{} \{U\llrra \bar U \}$ it holds\\
% $ \xymatrix{ { } & \{U\ra\bar U\} \ar[d] \\
%{X} \ar[r]_{\!\!\!\!\!\!\!\!\!\xi_U} \ar@{ -->}[ur] & { \{U\llrra \bar U \} }}$
%\ \ \ \ iff for each $\{u\}\lra X$, \ \ \ \ \ \ \
% $ \xymatrix{ {\{u\}} \ar[r] \ar[d] & \{u \ra U \llrra \bar U\} \ar[d] \\
%{X} \ar[r]_{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\xi_U} \ar@{ -->}[ur] & {
%\{u=U\llrra \bar U \} }}$
%\end{itemize}
%
_The preimage of an open set is open_ corresponds to the composition
X-->Y-->{u->u'}-->{u<->u'} .
This observation suggests that some arguments in elementary topology may be
understood entirely in terms of diagram chasing, see [Gavrilovich, Elementary Topology] for some
examples. This reinterpretation may help clarify the nature of the axioms of a
topological space, in particular it offers a constructive approach, may clarify
to what extent set-theoretic language is necessary, and perhaps help to suggest
an approach to ''tame topology'' of Grothendieck, Does this lead to tame
topology of Grothendieck, i.e. a foundation of topology "without false
problems" and "wild phenomena" "at the very beginning" ?
Let us state two problems; we hope they help to clarify the notion
of an ergosystem and that of a topological space.
_Problem_ 1. Write a short program which extracts diagram chasing derivations
from texts on elementary topology, in the spirit of the ideology of
ergosystems/ergostructures.
_Problem 2._ Develop elementary topology in terms of finite categories (viewed
as finite topological spaces) and labelled commutative diagrams, with an
emphasis on labels (properties) of morphisms defined by the Quillen lifting
property. Does this lead to tame topology of Grothendieck, i.e. a foundation
of topology "without false problems" and "wild phenomena" "at the very
beginning" ?
_Problem 3_. (Ganesalingam, Gowers) wrote an automatic theorem prover trying to
make it "thinking in a human way". In a couple of their examples their
generated proofs amount to diagram chasing, e.g. Claim (v) shows the generated
proof of the claim that a closed subspace of a complete metric space is
necessarily complete translates to two applications of a diagram chasing rule
corresponding to the lifting property, see [Gavrilovich, Slides] for details.
Arguably, both examples on the first page of (ibid.) also correspond to diagram
chasing. In their approach, can this be seen as evidence that humans are
really thinking by diagram chasing and the lifting property in particular? Note
that Clim (v) involves examples typically shown to students to clarify the
concepts of a metric space being complete or closed. Can one base a similar
theorem prover on our observations, particularly in elementary topology?
Let us comment on an approach to Problem 1.
We observed that there is a simple rule which leads to several notions in
topology interesting enough to be introduced in an elementary course. Can this
rule be extended to a very short program which learns elementary topology?
We suggest the following naive approach is worth thinking about.
The program maintains a collection of directed labelled graphs and certain
distinguished subgraphs. Directed graphs represent parts of a category;
distinguished subgraphs represent commutative diagrams. Labels represent
properties of morphisms. Further, the program maintains a collection of rules
to manipulate these data, e.g. to add or remove arrows and labels.
The program interacts with a flow of signals, say the text
of [Bourbaki, General Topology, Ch.1], and seeks correlations
between the diagram chasing rules and the flow of signals.
It finds a "correlation" iff certain strings occur nearby
in the signal flow iff they occur nearby in a diagram chasing rule.
To find "what's interesting", by brute force it searches
for a valid derivation which exhibits such correlations. To guide the search
and exhibit missing correlations in a derivation under consideration, it may
ask questions: are these two strings related? Once it finds such a derivation,
the program "uses it for building its own structure". Labels correspond to
properties of morphisms. Labels defined by the lifting property play an
important role, often used to exclude counterexamples making a diagram chasing
argument fail. In [DeMorgan] we analysed the text of the definitions of
surjective and injective maps showing what such a correlation may look like in
a "baby" case.
A related but easier task is to write a theorem prover doing diagram chasing
in a model category. The axioms of a (closed or not) model category
as stated in [Quillen,I.1.1] can be interpreted as rules to manipulate
labelled commutative diagrams in a labelled category. It appears
straightforward how to formulate a logic (proof system) based on these rules
which would allow to express statements like: Given a labelled commutative diagram,
(it is permissible to) add this or that arrow or label.
Moreover, it appears not hard to write a theorem prover for this logic
doing brute force guided search. What is not clear whether this logic is complete
in any sense or whether there are non-trivial inferences of this form to prove.
Writing such a theorem prover is particularly easy when the underlying category
of the model category is a partial order [GavrilovichHasson] and [BaysQuilder] wrote
some code for doing diagram chasing in such category. However, the latter
problem is particularly severe as well.
The two problems are related; we hope they help to clarify the notion
of an ergosystem and that of a topological space.
The following is a more concrete question towards Problem 2.
_Problem 2'_.
(a) Prove that a compact Hausdorff space is normal by diagram chasing;
does it require additional axioms?
Note that we know how to express the statement entirely in terms of the
lifting property and finite topological spaces of small size.
(b) Formalise the argument in [Fox, 1945] which implies the category of
topological spaces is not Cartesian closed. Namely, Theorem 3 [ibid.] proves that
if X is separable metrizable space, R is the real line, then
X is locally compact
iff
there is a topology on X^R such that for any space T,
a function h:XxT-->R is continuous iff
the corresponding function h*:T-->X^R is continuous
(where h(x,t)=h*(t)(x))
Note that here we do not know how to express the statement.
Remark. In [Gavrilovich, Tame Topology,\S6] and [Gavrilovich, Elementary Topology,\S2.9] we note that
uniform spaces may be viewed as simplicial objects in
the category of topological spaces in the following way.
We wish to emphasise that this observation can be easily "read off"
from (Bourbaki, General Topology, II\S1.1.1) if one is inclined
to translate everything into diagram chasing.
Let X be a set. Let
X<=>XxX<==>XxXxX<==>....
be the "trivial" simplicial set where degeneracy and face maps are
coordinate projections and diagonal embeddings.
A uniform structure on a set X is a filter, hence a topology, on the set XxX
satisfying certain properties; equip XxX with this topology.
Put the antidiscrete topology on X; put on XxXxX the topology which is
the pullback of the topologies on XxX and X along the projection maps, and
similarly for XxXxXxX etc. A straightforward verification shows this is well-defined
whenever the topology on XxX corresponds to a uniform structure on X.
Remark. In [Gavrilovich, Tame Topology, \S5.4] we observe that a number of consequences
of compactness can be expressed as a change of order of quantifiers in a
formula,
i.e. are of form AE \phi(..) ==> EA \phi (..) ,
namely that a real-valued function on a compact is necessarily bounded,
that a Hausdorff compact is necessarily normal,
that the image in X of a closed subset in XxK is necessarily closed,
Lebesgue's number Lemma, and paracompactness.
Such formulae correspond to inference rules of a special form,
and we feel a special syntax should be introduced to state
these rules.
For example, consider the statement that
"a real-valued function on a compact domain is necessarily bounded".
As a first order formula, it is expressed as
for all x in K there exists M ( f(x) <= M ) ==> there exists M for all x in K ( f(x) <= M )
Another way to express it is:
there exists M:K-->R for all x in K ( f(x) <= M(x) ) ==> there exists M in R for all x in K ( f(x) <= M )
Note that all that happened here is that a function M:K-->R,
become a constant M in R, or rather
expression "M(x)" of type K-->M which used (depended upon) variable "x"
become expression "M" which does not use (depend upon) variable "x".
We feel there should be a special syntax which would allow
to express above as an inference rule _ removing dependency of M(x) on x _,
and this syntax should be used to express consequences of compactness
in a diagram chasing derivation system for elementary topology.
To summarise, we think that compactness should be formulated
with help of inference rules for expressly manipulating which variables are 'new'
and what variables terms depend on, e.g. rules replacing a term t(x,y) by term t(x).
Something like the following:
... f(x) =< M(x) ...
_______________________
... f(x) =< M ...
## Ergo-Structures/Ergo-Systems Conjecture of Misha Gromov.
We conclude with a section which aims to explain our motivation and
hence is speculative and perhaps somewhat inappropriate in what is mostly
a mathematical text.
Misha Gromov [Memorandum Ergo] conjectures
there are particular mathematical, essentially combinatorial, structures,
called _ergostructures_ or _ergosystems_,
which help to understand complex biological behaviour including learning
and create mathematically interesting models of these processes.
We hope our observations may eventually help to uncover
an essentially combinatorial reasoning behind elementary topology,
and thereby suggest an example of an ergostructure.
An ergosystem/ergostructure may be thought of as an "engine" producing
(structurally or mathematically) _interesting_ behaviour
which is then misappropriated into a _useful_ behaviour by a biological
system.
"Behaviour" is thought of as an interaction with a flow of signals.
By itself, such an "engine" produces _interesting_ behaviour,
with little or no concern for any later use;
it "interacts with an incoming flow of signals;
it recognizes and selects what is _interesting_ for itself
in such a flow and uses it for building its own structure"
[Gromov, Memorandum Ergo].
An analogy might help. Consider a complex mechanical contraption powered by an
engine, such as a loom. By itself, there is nothing directly useful done by its
engine; indeed, the very same engine may be used by different mechanisms for
all sorts of useful and useless tasks. To understand the workings of a
mechanism, sometimes you had better forget its purpose and ask what is the
engine, how is it powered, and what keeps the engine in good condition.
Understanding the loom's engine (only) in terms of how it helps to weave is
misleading.
Thus, the concept of an ergostructure/ergosystems suggests a different kind of
questions we should ask about biological systems and learning.
A further suggestion is that these "engines" might be rather universal, i.e.
able to behave interestingly interacting with a diverse range of signal flows.
At the very beginning an ergosystem/structure is a "crisp" mathematically
interesting structure of size small enough to be stumbled upon by evolution; as
it grows, it becomes "fuzzy" and specialised to a particular kind of flow of
signals.
However, we want to draw attention to the following specific suggestion:
""The category/functor modulated structures can not be directly used by ergosystems,
e.g. because the morphisms sets between even moderate objects are
usually unlistable.
But the ideas of the category theory show that there are certain (often
non-obviuos) rules for generating proper concepts. (You ergobrain would not
function if it had followed the motto: "in my theory I use whichever definitions
I like".) The category theory provides a (rough at this stage) hint on a possible
nature of such rules.
[Gromov, Ergobrain]
Our observations give an example of a simple rule which can be used "to
generate proper concepts", particularly in elementary topology. We hope that
our observations can make the hint less rough, particularly if one properly
develops elementary topology in terms of diagram chasing, with an emphasis on
the lifting property.
_Problem_ 1. Write a short program which extracts diagram chasing derivations
from texts on elementary topology, in the spirit of the ideology of
ergosystems/ergostructures. That is, it considers a flow of signals
interesting if it correlates with diagram chasing rules.
In the last section we give some suggestions, albeit naive, what such a program
might look like.
_Problem 2._ Develop elementary topology in terms of labelled commutative
diagrams involving finite categories (viewed as finite topological spaces),
with an emphasis on labels (properties) of morphisms defined by the Quillen
lifting property.
Does this lead to the tame topology of Grothendieck, i.e. a foundation of topology
"without false problems" and "wild phenomena" "at the very beginning" ?
In the last section we give some suggestions, albeit naive, what such
a program might look like and how to express elementary topology
in terms of labelled diagram chasing.
Acknowledgements. To be written.
This work is a continuation of [DMG]; early history is given there. I thank
M.Bays, K.Pimenov, V.Sosnilo and S.Synchuk for discussions and proofreading; I
thank L.Beklemishev, N.Durov, S.V.Ivanov, A.L.Smirnov for discussions. I also
thank several students for encouraging and helpful discussions. Chebyshev
laboratory, St.Petersburg State University, provided a coffee machine and an
excellent company around it to chat about mathematics. Special thanks are to
Martin Bays for many corrections and helpful discussions. Several observations
in this paper are due to Martin Bays. I thank S.V.Ivanov for several
encouraging and useful discussions; in particular, he suggested to look at the
Lebesque's number lemma and the Arzela-Ascoli theorem. A discussion with Sergei
Kryzhevich motivated the group theory examples.
Much of this paper was done in St.Petersburg; it wouldn't have been possible
without support of family and friends who created an excellent social
environment and who occasionally accepted an invitation for a walk or a coffee
or extended an invitation; alas, I made such a poor use of it all.
This note is elementary, and it was embarrassing and boring, and
embarrassingly boring, to think or talk about matters so trivial, but luckily
I had no obligations for a time.
References:
[Alexandroff] Alexandroff, P. S.
Results in Topology in the last 25 years, Russian Math. Surveys, 15 (1960)
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=6719&option_lang=eng
[BaysQuilder]
Martin Bays.
A diagram chaser. Abandoned code for diagram chasing in a posetal model category.
http://mishap.sdf.org/quilder/ReadMe
[Bourbaki]
Nicolas Bourbaki.
General Topology. I\S10.2, Thm.1(d), p.101 (p.106 of file)
http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu
[Gavrilovich, Elementary Topology] Misha Gavrilovich.
Elementary general topology as diagram chasing calculations with finite categories. A draft of a research proposal.
http://mishap.sdf.org/by:gavrilovich/mints-elementary-topology-as-diagram-chasing.pdf
[Gavrilovich, Tame Topology] Misha Gavrilovich.
Tame topology: a naive elementary approach via finite topological spaces. A draft of a research proposal.
http://mishap.sdf.org/by:gavrilovich/zhenya-tame-topology.pdf
[Gavrilovich, DMG]
Misha Gavrilovich,
Point set topology as diagram chasing computations. Lifting properties as intances of negation.
The De Morgan Gazette 5 no.~4 (2014), 23--32, ISSN 2053-1451
http://mishap.sdf.org/by:gavrilovich/mints-lifting-property-as-negation-DMG_5_no_4_2014.pdf
[GavrilovichHasson]
Misha Gavrilovich, Assaf Hasson.
Exercises de style: A homotopy theory for set theory. Part I and II.
http://mishap.sdf.org/by:gavrilovich-and-hasson/what:a-homotopy-theory-for-set-theory/Exercises_de_style_A_homotopy_theory_for_set_theory-I-II-IJM.pdf
[GLZ]
Misha Gavrilovich, Alexandre Luzgarev, Vladimir Sosnilo.
A decidable fragment of diagram chasing without automorphisms.
preprint.
http://mishap.sdf.org/mints-a-decidable-fragment-of-category-theory-without-automorphisms.pdf
[Gromov, Ergobrain]
Misha Gromov.
Structures, Learning and Ergosystems: Chapters 1-4, 6.
December 30, 2011.
http://www.ihes.fr/~gromov/PDF/ergobrain.pdf
[Gromov, Memorandum Ergo]
Misha Gromov.
Memorandon Ergo. October 29, 2015.
http://www.ihes.fr/~gromov/PDF/ergo-cut-copyOct29.pdf
Russian translation in Gromov M., Kolco tain: vselennaya, matematika, mysl'.
Per. s angl.yaz. N.B.Tsilevich. M.: MCNMO, 2017. 289 c. ISBN 978-5-4439-1117-5.
[Taimanov] A. D. Taimanov, "On extension of continuous mappings of topological spaces", Mat. Sb. (N.S.), 31(73):2 (1952), 459-463
## Appendix A. Separation axioms as lifting properties
The separation axioms are lifting properties with respect to maps involving up to 4 points
and the real line.
In all of the following definitions, X is again a topological space.
X is T0, or Kolmogorov, if any two distinct points in X are topologically
distinguishable. (It will be a common theme among the separation axioms to have
one version of an axiom that requires T0 and one version that doesn't.)
{x<->y} ---> {x=y} /_ X --> {o}
X is R0, or symmetric, if any two topologically distinguishable points in X
are separated.
{x->y} ---> {x<->y} /_ X ---> {o}
X is T1, or accessible or Fréchet, if any two distinct points in X are
separated. Thus, X is T1 if and only if it is both T0 and R0. (Although you may
say such things as "T1 space", "Fréchet topology", and "Suppose that the
topological space X is Fréchet", avoid saying "Fréchet space" in this
context, since there is another entirely different notion of Fréchet space in
functional analysis.)
{x->y} ---> {x=y} /_ X ---> {o}
X is R1, or preregular, if any two topologically distinguishable points in
X are separated by neighbourhoods. Every R1 space is also R0.
X is weak Hausdorff, if the image of every continuous map from a compact
Hausdorff space into X is closed. All weak Hausdorff spaces are T1, and all
Hausdorff spaces are weak Hausdorff.
X is Hausdorff, or T2 or separated, if any two distinct points in X are
separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T0
and R1. Every Hausdorff space is also T1.
{x,y} >--> X /_ {x->X<-y} --> {x=X=y}
X is T2½, or Urysohn, if any two distinct points in X are separated by
closed neighbourhoods. Every T2½ space is also Hausdorff.
{x,y} >--> X /_ {x->x'<-X->y'<-y} --> {x=x'=X=y'=y}
X is completely Hausdorff, or completely T2, if any two distinct points in
X are separated by a continuous function. Every completely Hausdorff space is
also T2½.
{x,y} >--> X /_ [0,1]-->{o}
here {x,y} >--> X runs through all injective maps from the discrete two
point space {x,y}
X is regular if, given any point x and closed set F in X such that x does
not belong to F, they are separated by neighbourhoods. (In fact, in a regular
space, any such x and F will also be separated by closed neighbourhoods.) Every
regular space is also R1.
{x} ---> X /_ {x->X<-F'->F} ---> {x=X=F'->F}
X is regular Hausdorff, or T3, if it is both T0 and regular.[1] Every
regular Hausdorff space is also T2½.
X is completely regular if, given any point x and closed set F in X such
that x does not belong to F, they are separated by a continuous function. Every
completely regular space is also regular.
{x} ---> X /_ [0,1] \union {F} ---> {x->F}
where points F and 1 are topologically indistinguishable, [0,1] goes to x,
and F goes to F.
X is Tychonoff, or T3½, completely T3, or completely regular Hausdorff, if
it is both T0 and completely regular.[2] Every Tychonoff space is both regular
Hausdorff and completely Hausdorff.
X is normal if any two disjoint closed subsets of X are separated by
neighbourhoods. (In fact, a space is normal if and only if any two disjoint
closed sets can be separated by a continuous function; this is Urysohn's lemma.)
{} --->X /_ {x<-x'->X<-y'->y} ---> {x<-x'=X=y'->y}
{} ---> X /_ {0'} \union [0,1] \union {1'} ---> {0=0'->x<-1=1'} where points
0',0 and 1,1' are topologically indistinguishable,
[0,1] goes to x, and 0,0' map to point 0=0', and 1,1' map to point
1=1'.
X is normal Hausdorff, or T4, if it is both T1 and normal. Every normal
Hausdorff space is both Tychonoff and normal regular.
X is completely normal if any two separated sets A and B are separated by
neighbourhoods U=)A and V=)B such that U and V do not intersect.
Every completely normal space is also normal.
{} ---> X /_ {x<-A<->U->U'<-W->V'<-V<->B->x} ---> {U=U',V'=V}
X is perfectly normal if any two disjoint closed sets are precisely
separated by a continuous function. Every perfectly normal space is also
completely normal.
{}-->X /_ [0,1]-->{0<-X->1}
where (0,1) goes to the open point X, and 0 goes to 0, and 1 goes to 1.