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- #1

- Feb 15, 2012

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De-mystifying universal mapping properties: an example-quotient groups.

- Thread starter Deveno
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- Thread starter
- #1

- Feb 15, 2012

- 1,967

De-mystifying universal mapping properties: an example-quotient groups.

- Jun 22, 2012

- 2,891

De-mystifying universal mapping properties: an example-quotient groups.

Hi Deveno ... thank you for posting on this topic ... so helpful as the topic is certainly mystifying and highly unsettling ...

Indeed ... you write ...

"Often, in the study of algebraic objects certain things (like tensor products) are often defined primarily in terms of an universal mapping property. When one is used to "concrete objects" one can calculate with, this often comes as a shock to the system. One feels as if one is spinning something out of nothing, and has the feeling of "waiting for the real 'stuff' to show up." ... ... "

Exactly!!! You are right on the money ... ... precisely how I feel when approaching this material ... ... but on the other hand ... it is quite fascinating ... as the spider's web is to the fly ... ...

Now ... back to studying your posts ... ...

Peter

- Jun 22, 2012

- 2,891

De-mystifying universal mapping properties: an example-quotient groups.

You write:

View attachment 5587

In the above definition A can be any set ... so don't we have to show that Definition 1 implies Definition 2 for any ... and hence all sets ...?

Can you clarify this matter ... ?

Peter

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- #4

- Feb 15, 2012

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That would be the second half of the proof. I am proving Definition 2 implies Definition 1, here. Since Definition 2 is quite general, we lose nothing by using some particular instance of it.You write:

In the above definition A can be any set ... so don't we have to show that Definition 1 implies Definition 2 for any ... and hence all sets ...?

Can you clarify this matter ... ?

Peter

- Jun 22, 2012

- 2,891

That would be the second half of the proof. I am proving Definition 2 implies Definition 1, here. Since Definition 2 is quite general, we lose nothing by using some particular instance of it.

Oh ... OK ... yes, of course ... thanks for pointing that out ...

Peter

Last edited:

- Jun 22, 2012

- 2,891

De-mystifying universal mapping properties: an example-quotient groups.

This series of posts continues to be just amazingly helpful ...

These posts cover what is to me an unfamiliar and difficult area with a clarity that seems to me to be completely lacking in the standard texts that seem to assume you already have a basic understanding of UMPs ...

So ... THANK YOU ...

Peter