abstraction of notation from sets.

abstraction of notation from sets.

[peasthope]

When S is a set, the notation "a \epsilon S" is familiar.
Is this ever extended to CT?  All the texts I recall use
natural language such as "A is an object of C".  What if
a more symbolic notation is required?

Thanks,       ... Peter E.


-- 
Google "pathology workshop"



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Re: abstraction of notation from sets.

[Johannes Huebschmann]

Dear All

"a \epsilon S" is part of the language
(ZF).

"A is an object of C" is metalanguage.

Unfortuantely this gets often confused.

I have still known Bernays in my mathematical youth.
He would have strongly objected to write
"A \epsilon C" for
"A is an object of C".

Best Johannes


On Tue, 23 Feb 2010, peasthope@shaw.ca wrote:

> When S is a set, the notation "a \epsilon S" is familiar.
> Is this ever extended to CT?  All the texts I recall use
> natural language such as "A is an object of C".  What if
> a more symbolic notation is required?
>
> Thanks,       ... Peter E.
>
>
>


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Re: abstraction of notation from sets.

[Mikael Vejdemo-Johansson]

On Feb 23, 2010, at 4:43 PM, peasthope@shaw.ca wrote:
> When S is a set, the notation "a \epsilon S" is familiar.
> Is this ever extended to CT?  All the texts I recall use
> natural language such as "A is an object of C".  What if
> a more symbolic notation is required?
>

I've seen $a \in Ob(C)$ numerous times, and also - though primarily
from Barr & Wells - $a \in C_0$, with the rationale that a category is
a graph (consisting of vertices C_0 and edges C_1), with extra
conditions introduced to capture the composition operation, showing up
as functions defined on composable sequences C_n  of n edges (most
often for n=2, or 3 for associativity).

-- Mikael Vejdemo Johansson


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Re: abstraction of notation from sets.

[Aleks Kissinger]

\epsilon is often used informally to mean an object or an arrow is contained
in a category C. I.e. a \in C is shorthand for a \in ob(C) or a \in ar(C),
when its interpretation is obvious from context.

Barr and Wells (Toposes, Triples and Theories, 1984) uses set inclusion
notation in an interesting way, thinking of arrows rather as "generalised
elements" of objections. So, they'll write something like f \in^A B instead
of f : A -> B, which should be read "f is an A-element of B". When the
category is sets and A = {*}, this is the usual notion of element. This
notion gives the right kind of intuition when working and categories that
are a lot like sets, especially toposes. The language also admits a
particularly beautiful way to express the Yoneda lemma.

"The (normal) elements of FA are the same as the hom(-,A)-elements of F."

a

On Wed, Feb 24, 2010 at 12:43 AM, <peasthope@shaw.ca> wrote:

> When S is a set, the notation "a \epsilon S" is familiar.
> Is this ever extended to CT?  All the texts I recall use
> natural language such as "A is an object of C".  What if
> a more symbolic notation is required?
>
> Thanks,       ... Peter E.
>
>

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Re: abstraction of notation from sets.

[Partha Pratim Ghosh]

 <peasthope <at> shaw.ca> writes:

>
> When S is a set, the notation "a \epsilon S" is familiar.
> Is this ever extended to CT?  All the texts I recall use
> natural language such as "A is an object of C".  What if
> a more symbolic notation is required?
>
> Thanks,       ... Peter E.
>

Dear Peter,

   There are two types of entities, one called objects and the other called
arrows. However, there are ways to deal away with objects, and one could only
consider the arrows, in which case although in my opinion the presentation
becomes much more formal and less intuitive (?!), one could easily import the
symbols of $\epsilon$; alternatively, one could formulate two classes (one for
objects and other for arrows) and do a similar import; or else one could use a
type theoretic fashion, say "A.Obj" to denote an instance of objects and
"A.Arr" to denote an instance of arrows, and so on. Yet, in some form or the
other one could do this...., but I ponder: why?

   Thus, I could not quite understand the intent of your question. Please could
you elaborate on this.

   With my regards,

partha




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Re: abstraction of notation from sets.

[Michael Shulman]

<peasthope <at> shaw.ca> writes:
> When S is a set, the notation "a \epsilon S" is familiar.
> Is this ever extended to CT?  All the texts I recall use
> natural language such as "A is an object of C".  What if
> a more symbolic notation is required?

As has been pointed out, if category theory is formalized within set
theory, so that a category has a set of objects and a set of arrows,
then one cannot write "a \in C" to mean that a is an object of the
category C, at least as long as \in is restricted to its precise
set-theoretic meaning.  However, in my experience it is fairly common to
write "a \in C" with this meaning, although perhaps not so common in
formal mathematical writing.

I regard this as precisely analogous to writing "g \in G" when G is a
group, since after all when formalized within set theory, a group is not
just the set of its elements, but a triple (G,m,e) of a set, a
multiplication, and an identity (or some other equivalent encoding).
One can refer to this sort of thing perjoratively as an "abuse of
notation," but one can also regard it as a perfectly legitimate part of
informal mathematical language which is not captured by the
set-theoretic encoding.  One could also formalize it by regarding the
symbol "\in" as "overloaded" in a precise sense analogous to programming
languages.

Mike


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Re: abstraction of notation from sets.

[Richard Garner]

> One cannot write "a \in C" to mean that a is an object of
> the category C, at least as long as \in is restricted to
> its precise set-theoretic meaning.  However, in my
> experience it is fairly common to write "a \in C" with this
> meaning, although perhaps not so common in formal
> mathematical writing.

> One can refer to this sort of thing perjoratively as an "abuse of
> notation," but one can also regard it as a perfectly legitimate part of
> informal mathematical language which is not captured by the
> set-theoretic encoding.  One could also formalize it by regarding the
> symbol "\in" as "overloaded" in a precise sense analogous to programming
> languages.

I think it is more perspicuous to treat this, not as an
overloading of \in, but as an "implicit conversion"
associated to the notion of category; that is, we allow the
forgetful functor Cat->Set to be applied silently in contexts
which otherwise would not type-check. In fact the vast
majority of "abuses of notation" are of this character, when
applied to, for example, any forgetful functor from an
Eilenberg-Moore category; the discrete category functor
Set->Cat; the Yoneda embedding C -> [C^op, Set]; the
forgetful functor from universal cones to their vertex, etc,
etc. In principle this becomes problematic as soon as the
category generated by all such implicit conversions has
non-identity idempotents; in practice, this category is free
on a graph and we hope to identify a shortest path between
two vertices!

Richard


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Re: abstraction of notation from sets.

[Paul Levy]

On 26 Feb 2010, at 18:53, Richard Garner wrote:
>
> I think it is more perspicuous to treat this, not as an
> overloading of \in, but as an "implicit conversion"
> associated to the notion of category; that is, we allow the
> forgetful functor Cat->Set to be applied silently in contexts
> which otherwise would not type-check. In fact the vast
> majority of "abuses of notation" are of this character, when
> applied to, for example, any forgetful functor from an
> Eilenberg-Moore category; the discrete category functor
> Set->Cat; the Yoneda embedding C -> [C^op, Set]; the
> forgetful functor from universal cones to their vertex, etc,
> etc. In principle this becomes problematic as soon as the
> category generated by all such implicit conversions has
> non-identity idempotents;

Why restrict this to idempotents?  Surely the category needs to be a
preorder for the usage to be unambiguous?

Paul


> in practice, this category is free
> on a graph and we hope to identify a shortest path between
> two vertices!
>
> Richard
>


--
Paul Blain Levy
School of Computer Science, University of Birmingham
+44 (0)121 414 4792
http://www.cs.bham.ac.uk/~pbl











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Re: abstraction of notation from sets.

[Vaughan Pratt]

Besides Peter Easthope's question on the appropriateness of considering
an object of C as an element of C there is also (in fact more generally
as I'll mention near the end) the question of whether the notion of
"element" is well established in a presheaf category C = Set^{J^op} on
J, made a topos by choice of a final object 1 and a subobject classifier
O, and (by Yoneda) with the further choice of a full embedding of J in
C.  With that arrangement, and with the idea that distinct provenances
for sets give rise to distinct notions of "element," it seems to me that
at least four natural kinds of set arise in a presheaf topos in ordinary
mathematical practice.

First kind.  As a homset from 1.  This notion makes the connection with
set theory that toposes were developed for, with the caveat that it
should be understood in the light of the third and fourth kinds so as
not to overstate its applicability.

Second kind.  As a morphism to O.  This represents a subobject of the
domain of the morphism (the subobject itself being in general a proper
class and therefore not a fit entity for ordinary mathematics).  For
example in the topos Set with natural numbers object N it is natural to
write {2,3,5,7} for the set of 1-digit primes, understood as (the
subobject of N represented by) its characteristic function as a morphism
from N to O.

Third kind.  As a homset to O.  This is the power *set* C(X,O) of
subobjects of the domain X of the homset, which by the cartesian closed
structure of C is in a natural bijection with the power *object* O^X
when considered as a set C(1,O^X) of the first kind.

Fourth kind.  As a homset from the image Y(j) under the embedding Y: J
--> C of some object j of J.  For example if J has objects V and E
making each object of the topos a graph G then we think of G as formed
from two sets, and refer to the morphisms to G from Y(V) and Y(E) as
respectively the vertices and edges of G.  In this example the set of
vertices of G also happens to be a set of the first kind, but not the
set of edges, pointing up the need I mentioned earlier not to overstate
the significance of sets of the first kind while also addressing Peter's
original question in a roundabout way in terms of the underlying graph
of a category.  A morphism of a presheaf category is properly understood
as an ob(J)-indexed family of functions mapping sets of the fourth kind
to their counterparts in the codomain of the morphism.  Among these the
monics as families of injections furnish the notion of subobject in a
presheaf topos with its intuitive meaning complementary to that of the
second kind of set, the remark about proper classes notwithstanding.

Of course any homset of a topos is a set, but it seems to me that the
above four kinds deserve special recognition as sets commonly
encountered in mathematical practice having readily distinguishable
provenances.

Vaughan Pratt


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Re: abstraction of notation from sets.

[Toby Bartels]

In a context where "C" is a term for a category and "a" is a new variable,
I would interpret "a \in C" to declare that a is an object of C,
but I wouldn't be surprised if the author really meant to declare
that a is a map in C (with notation for source and target derived).
Using scripts can make things more clear, as peasthope@shaw.ca wrote:

>"$a \in C_0$" = "a is an object in C"
>"$f \in C_1$" = "f is a map in C"

This generalises immediately to higher categories.

If "a" is not a new variable but instead a term with an established meaning,
then "a \in C" might mean either that a is an object or a map of C,
but the meaning should be clear from context, particularly if "C" is
a term for a subcategory of a category D in which a is an object or map.

Even in set theory, there can be a logical difference between
declaring that a new variable stands for an element of a given set
and stating that a given element belongs to a given subset,
and people using structural or type-theoretic approaches to set theory
sometimes even use different notation to distinguish these.
One possibility (not the only one) is to use a colon for the former;
then if "a" is a new variable and "S" is a term for a set,
then "a: S" would declare that a is an element of S;
while if "a" is a term for an element of some set T
and "S" is a term for a subset of T, then "a \in S"
would denote the proposition that a belongs to S.
The former ("a: S") only makes sense as an assertion or hypothesis,
but the latter ("a \in S") makes sense in any logical context.
(In particular, it makes sense to say "If a \notin S, then ..."
but not "If a: S is false, then ...".)

This can be extended to category theory as follows:
If "a" is a new variable and "C" is a term for a category,
then let "a: C" declare that a is an object of C.
If "f" is a new variable and "a" and "b" are terms for objects of C,
then let "f: a \to b" declare that f is a map in C from a to b,
which after all is already a very widely used notation.
(We can also write "f: a \to b: C" to declare everything at once.)
This generalises to higher categories, as long as you follow the plan
of naming sources and targets before you name whatever goes between them.
(If you follow the philosophy of avoiding "evil", then this is very natural.)
On the other hand, if D is a category and "C" is a term for a subcategory,
then "a \in C" has an unambiguous meaning as long as "a" is a term
for an element at some level (object or map) in D;
this also generalises to higher categories.


--Toby


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Re: abstraction of notation from sets.

[peasthope]

Mikael,

> ... $a \in C_0$, with the rationale that a category is
> a graph (consisting of vertices C_0 and edges C_1), ...

So
"$a \in C_0$" = "a is an object in C"
and
"$f \in C_1$" = "f is a map in C"
would be acceptable to some readers?

Thanks,         ... Peter E.





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