Comments on: Regular and effective monomorphisms and epimorphisms
https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/
"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul HalmosMon, 18 May 2015 03:54:34 +0000
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By: Generators | Annoying Precision
https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/#comment-5100
Mon, 18 May 2015 03:54:34 +0000http://qchu.wordpress.com/?p=11021#comment-5100[…] a category won’t have fake isos if its epis are well-behaved; for example, whenever epis are regular, a hypothesis we’ll use later as well. This holds in abelian categories but also in and ; […]
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By: Tiny objects | Annoying Precision
https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/#comment-5084
Thu, 07 May 2015 21:26:08 +0000http://qchu.wordpress.com/?p=11021#comment-5084[…] is a coequalizer it is a regular epimorphism, and in particular an epimorphism, so it is right cancellative. Cancelling it […]
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By: Epi-mono factorizations « Annoying Precision
https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/#comment-2691
Tue, 13 Nov 2012 06:14:58 +0000http://qchu.wordpress.com/?p=11021#comment-2691[…] Today we will discuss some general properties of factorizations of a morphism into an epimorphism followed by a monomorphism, or epi-mono factorizations. The failure of such factorizations to be unique turns out to be closely related to the failure of epimorphisms or monomorphisms to be regular. […]
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By: Zhen Lin
https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/#comment-2563
Sun, 04 Nov 2012 08:24:50 +0000http://qchu.wordpress.com/?p=11021#comment-2563There’s an interesting corollary to the proposition that a morphism is a regular epimorphism if and only if it is the coequaliser of its kernel pair: assuming the axiom of choice, epimorphisms in Set are preserved by any functor whatsoever; this can then be used to show that the forgetful functor from the Eilenberg–Moore category of a monad over Set preserves coequalisers _of kernel pairs_, and hence, the _regular_ epimorphisms in the Eilenberg–Moore category are exactly those whose underlying map is a surjection.
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