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Localisation of Category

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Localisation of Category



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Category Theory with and without ObjectsWhat is a “foo” in category theory?Question on category theoryN-Tuples or N-functions in category theoryCan the choice of definition of morphisms for a slice category be justified categorically?Morphisms as zigzags, composition as concatenationSome doubts in Category TheoryCoslice category in 2-categoriesStatus of pairs/tuples in category theoryA new category $C^*$ from a given category $C$










2












$begingroup$


I have a quite broad question about localisations of categories:



Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?










share|cite|improve this question









$endgroup$











  • $begingroup$
    Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
    $endgroup$
    – Eric Wofsey
    2 hours ago















2












$begingroup$


I have a quite broad question about localisations of categories:



Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?










share|cite|improve this question









$endgroup$











  • $begingroup$
    Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
    $endgroup$
    – Eric Wofsey
    2 hours ago













2












2








2





$begingroup$


I have a quite broad question about localisations of categories:



Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?










share|cite|improve this question









$endgroup$




I have a quite broad question about localisations of categories:



Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?







category-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









KarlPeterKarlPeter

7411416




7411416











  • $begingroup$
    Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
    $endgroup$
    – Eric Wofsey
    2 hours ago
















  • $begingroup$
    Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
    $endgroup$
    – Eric Wofsey
    2 hours ago















$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago




$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago










1 Answer
1






active

oldest

votes


















7












$begingroup$

If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
    $endgroup$
    – Eric Wofsey
    2 hours ago










  • $begingroup$
    @EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
    $endgroup$
    – Kevin Carlson
    2 hours ago











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









7












$begingroup$

If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
    $endgroup$
    – Eric Wofsey
    2 hours ago










  • $begingroup$
    @EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
    $endgroup$
    – Kevin Carlson
    2 hours ago















7












$begingroup$

If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
    $endgroup$
    – Eric Wofsey
    2 hours ago










  • $begingroup$
    @EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
    $endgroup$
    – Kevin Carlson
    2 hours ago













7












7








7





$begingroup$

If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.






share|cite|improve this answer









$endgroup$



If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 hours ago









Kevin CarlsonKevin Carlson

34k23473




34k23473











  • $begingroup$
    For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
    $endgroup$
    – Eric Wofsey
    2 hours ago










  • $begingroup$
    @EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
    $endgroup$
    – Kevin Carlson
    2 hours ago
















  • $begingroup$
    For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
    $endgroup$
    – Eric Wofsey
    2 hours ago










  • $begingroup$
    @EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
    $endgroup$
    – Kevin Carlson
    2 hours ago















$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago




$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago












$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago




$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago

















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