Localisation of Category Announcing the arrival of Valued Associate #679: Cesar Manara ...
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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 $C$
$begingroup$
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?
category-theory
asked 9 hours ago
KarlPeterKarlPeter
7411416
$endgroup$
add a comment |
$begingroup$
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?
category-theory
asked 9 hours ago
KarlPeterKarlPeter
7411416
$endgroup$
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
9 hours ago
add a comment |
$begingroup$
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?
category-theory
asked 9 hours ago
KarlPeterKarlPeter
7411416
$endgroup$
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
category-theory
asked 9 hours ago
KarlPeterKarlPeter
7411416
asked 9 hours ago
KarlPeterKarlPeter
7411416
asked 9 hours ago
KarlPeterKarlPeter
7411416
asked 9 hours ago
KarlPeterKarlPeter
7411416
asked 9 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
9 hours ago
add a comment |
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
9 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
9 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
9 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^{-1}cd^{-1}...$ There's no way to simplify this into a traditional fraction-it need not be equal to $frac{ac}{b^{-1}d^{-1}}$, 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.
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
$begingroup$
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
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 hours ago
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^{-1}cd^{-1}...$ There's no way to simplify this into a traditional fraction-it need not be equal to $frac{ac}{b^{-1}d^{-1}}$, 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.
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
$begingroup$
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
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 hours ago
add a comment |
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^{-1}cd^{-1}...$ There's no way to simplify this into a traditional fraction-it need not be equal to $frac{ac}{b^{-1}d^{-1}}$, 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.
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
$begingroup$
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
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 hours ago
add a comment |
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^{-1}cd^{-1}...$ There's no way to simplify this into a traditional fraction-it need not be equal to $frac{ac}{b^{-1}d^{-1}}$, 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.
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^{-1}cd^{-1}...$ There's no way to simplify this into a traditional fraction-it need not be equal to $frac{ac}{b^{-1}d^{-1}}$, 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.
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
answered 9 hours ago
Kevin CarlsonKevin Carlson
34k23473
answered 9 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
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 hours ago
add a comment |
$begingroup$
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
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 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
9 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
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
9 hours ago
add a comment |
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$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
9 hours ago