Re: How to visualize limits in category theory?
From: Todd Trimble (trimble1_at_optonline.net)
Date: 12/20/04
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Date: Mon, 20 Dec 2004 14:23:24 +0000 (UTC)
On 20 Dec 2004, Jose Capco wrote:
>Thank you all for the nice tips. Let me add that when I discussed this
>with my supervisor, he said that it is nice to visualize limits in say
>the category of topologies as product of two spaces while colimits can
>be seen as the "joining" of two spaces (morphisms I suppose are just the
>homeomorphisms) in a pushout/pullback diagram (ie. a commutative
>rectangle such that if 3 objects and 2 morphisms are fixed and another
>commutative rectangle is formed with those fixed points and morphisms
>then there is a unique morphism from/to the 4th object of the old
>rectangle to/from the 4th object of the newly formed rectangle!)...
>
Well... this sounds like an extract from a conversation where
your supervisor was trying to impart some feeling for limits and
colimits through some *examples*, but it's stretching it to say
"visualize limits... as products" or "visualize colimits... as
pushouts". I *would* say that once you really understand the
universal property definition of things like (co)product,
(co)equalizer, pushout, pullback, etc., then the universality
property definition of general limits/colimits is a snap. So
it makes sense to master the simple examples first, which, yes,
are generally easy to visualize in concrete situations.
But sometimes such examples are *too* simple, too easy to glide
over. So you might want to wrap your head around the universal
property formulation of things like free products and amalgamated
products of groups (coproducts and pushouts, respectively),
where universal properties may seem more clarifying than in
the simpler examples. At an even greater remove in complexity
are schemes, and here being fluent with limits (fiber products
for instance) really pays off in the end, since the naive set-
theoretic intuitions no longer quite cut it here.
>Anyway, let me give some comments on one of the posts...
>
>Todd Trimble wrote:
>
>> You've gotten several good responses to your question; perhaps
>> what you need are examples to test the responses on.
>>
>> Jesse Hughes pointed out that limits are reducible to cartesian
>> products and equalizers, and James Dolan observed that limits
>> are the basic stuff of algebraic geometry as initiated by Fermat,
>> Descartes, and others (loci of equations seen as equalizers
>> defined on cartesian products).
>>
>> You might use these observations to visualize how the ring of
>> p-adic integers is constructed as a limit of a diagram of shape
>>
>> --> Z_{p^n} --> Z_{p^{n-1}} --> ... --> Z_p.
>
>This looks interesting. Thanks! Let me check only if I got it
>correctly... The limit would then be those integers which have the form
>p^k .. ie. Lim(Z_p) := {p^k : k is in N} , accompanied with the
>canonical homomorphisms to each of the Z_p^n ... right?
>
Sorry, but no. Denote an element of Z_{p^n} by
a_0 + a_1 p + ... + a_{n-1}p^{n-1}. If you work through
this example again, you should find that the elements in the
limit are similarly expressible as
a_0 + a_1 p + ... + a_ p^n + ...
(completion of Z w.r.t. p-adic metric, where p^n --> 0 as
n --> oo).
>
>
>> You might also visualize the patching condition in the definition
>> of sheaves F in terms of limits: if V is open and {V_i} covers V,
>> then F(V) is the limit of a diagram obtained by sticking together
>> diagrams of shape
>>
>> F(V_i) --> F(V_i cap F_j) <-- F(V_j)
>>
>> over all i and j.
>>
>
>Wow... very nice example. Er.. what does patching mean btw (is that one
>of the axioms of the sheaves?)
>
Yes; a sheaf is a presheaf which satisfies the patching condition:
given a cover {U_i} of U and sections s_i over U_i which
agree over intersections, there is a unique section s over U
which restricts to the s_i.
Todd Trimble
- Next message: BB: "Re: Is zero even or odd?"
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