Another Math Question
-
I don't see a math forum, so I'll tack on another math question here if anyone is familiar with abstract algebra and/or category theory. I've been learning category theory and have gotten stumped when learning about different characteristics of morphisms - specifically monomorphisms and epimorphisms. A monomorphism is defined as "a morphism f: X -> Y such that for all objects Z and all morphisms g1, g2: Z -> X, f o g1 = f o g2 implies g1 = g2." In English, it means that if we take a morphism f and compose it with any two morphisms "leading into" f, if the results are equal that implies the two morphisms are equal. It's basically the injective (or 1-to-1) property but for morphisms. An epimorphism is the same idea but for morphisms that f "leads into." It's basically the surjective (or onto) property but for morphisms. So here's where I'm confused: it's beaten into your head in textbooks to not think of categories and objects as concrete. The whole point is the morphisms. But how can you show characteristics like monomorphic or epimorphic behavior without analyzing the morphisms in a concrete object context? If you don't, how can you 1) guarantee the category is even equipped with the concept of equality? (not everything is a setoid), and 2) show that the equality holds per the mono- and/or epi-morphic definition? I hope that makes some sense. I'm still very much in the learning stage on this topic :-O
-
I don't see a math forum, so I'll tack on another math question here if anyone is familiar with abstract algebra and/or category theory. I've been learning category theory and have gotten stumped when learning about different characteristics of morphisms - specifically monomorphisms and epimorphisms. A monomorphism is defined as "a morphism f: X -> Y such that for all objects Z and all morphisms g1, g2: Z -> X, f o g1 = f o g2 implies g1 = g2." In English, it means that if we take a morphism f and compose it with any two morphisms "leading into" f, if the results are equal that implies the two morphisms are equal. It's basically the injective (or 1-to-1) property but for morphisms. An epimorphism is the same idea but for morphisms that f "leads into." It's basically the surjective (or onto) property but for morphisms. So here's where I'm confused: it's beaten into your head in textbooks to not think of categories and objects as concrete. The whole point is the morphisms. But how can you show characteristics like monomorphic or epimorphic behavior without analyzing the morphisms in a concrete object context? If you don't, how can you 1) guarantee the category is even equipped with the concept of equality? (not everything is a setoid), and 2) show that the equality holds per the mono- and/or epi-morphic definition? I hope that makes some sense. I'm still very much in the learning stage on this topic :-O
-
I don't see a math forum, so I'll tack on another math question here if anyone is familiar with abstract algebra and/or category theory. I've been learning category theory and have gotten stumped when learning about different characteristics of morphisms - specifically monomorphisms and epimorphisms. A monomorphism is defined as "a morphism f: X -> Y such that for all objects Z and all morphisms g1, g2: Z -> X, f o g1 = f o g2 implies g1 = g2." In English, it means that if we take a morphism f and compose it with any two morphisms "leading into" f, if the results are equal that implies the two morphisms are equal. It's basically the injective (or 1-to-1) property but for morphisms. An epimorphism is the same idea but for morphisms that f "leads into." It's basically the surjective (or onto) property but for morphisms. So here's where I'm confused: it's beaten into your head in textbooks to not think of categories and objects as concrete. The whole point is the morphisms. But how can you show characteristics like monomorphic or epimorphic behavior without analyzing the morphisms in a concrete object context? If you don't, how can you 1) guarantee the category is even equipped with the concept of equality? (not everything is a setoid), and 2) show that the equality holds per the mono- and/or epi-morphic definition? I hope that makes some sense. I'm still very much in the learning stage on this topic :-O
-
I don't see a math forum, so I'll tack on another math question here if anyone is familiar with abstract algebra and/or category theory. I've been learning category theory and have gotten stumped when learning about different characteristics of morphisms - specifically monomorphisms and epimorphisms. A monomorphism is defined as "a morphism f: X -> Y such that for all objects Z and all morphisms g1, g2: Z -> X, f o g1 = f o g2 implies g1 = g2." In English, it means that if we take a morphism f and compose it with any two morphisms "leading into" f, if the results are equal that implies the two morphisms are equal. It's basically the injective (or 1-to-1) property but for morphisms. An epimorphism is the same idea but for morphisms that f "leads into." It's basically the surjective (or onto) property but for morphisms. So here's where I'm confused: it's beaten into your head in textbooks to not think of categories and objects as concrete. The whole point is the morphisms. But how can you show characteristics like monomorphic or epimorphic behavior without analyzing the morphisms in a concrete object context? If you don't, how can you 1) guarantee the category is even equipped with the concept of equality? (not everything is a setoid), and 2) show that the equality holds per the mono- and/or epi-morphic definition? I hope that makes some sense. I'm still very much in the learning stage on this topic :-O
Practitioners of the black arts for fell purposes are not likely to receive help here. If we can't help the
gimme codezweinerschnotz, you shouldn't expect it either :-D .Software Zen:
delete this; -
That's morpheus, not morphism ;) :-D
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
-
That's morpheus, not morphism ;) :-D
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
-
Practitioners of the black arts for fell purposes are not likely to receive help here. If we can't help the
gimme codezweinerschnotz, you shouldn't expect it either :-D .Software Zen:
delete this;I'm kind of in no-man's land on this topic. I'm not a mathematician enough that answers I've found to similar-ish questions provide useful insight to this particular question, but it's also a question that I'm not sure many non-mathematicians would ask :(( I just got really interested in it because it's a fascinating branch of mathematics.
-
I don't see a math forum, so I'll tack on another math question here if anyone is familiar with abstract algebra and/or category theory. I've been learning category theory and have gotten stumped when learning about different characteristics of morphisms - specifically monomorphisms and epimorphisms. A monomorphism is defined as "a morphism f: X -> Y such that for all objects Z and all morphisms g1, g2: Z -> X, f o g1 = f o g2 implies g1 = g2." In English, it means that if we take a morphism f and compose it with any two morphisms "leading into" f, if the results are equal that implies the two morphisms are equal. It's basically the injective (or 1-to-1) property but for morphisms. An epimorphism is the same idea but for morphisms that f "leads into." It's basically the surjective (or onto) property but for morphisms. So here's where I'm confused: it's beaten into your head in textbooks to not think of categories and objects as concrete. The whole point is the morphisms. But how can you show characteristics like monomorphic or epimorphic behavior without analyzing the morphisms in a concrete object context? If you don't, how can you 1) guarantee the category is even equipped with the concept of equality? (not everything is a setoid), and 2) show that the equality holds per the mono- and/or epi-morphic definition? I hope that makes some sense. I'm still very much in the learning stage on this topic :-O
I would have to look up some translations to be precise, but it looks like your question is much more general than something about specific types of morphism: First of all, while Mathemtatics is mostly about the art to describe and solve problems separate from concrete examples, examples are still often used to illustrate corner cases, to disprove a false assumption, and occasionally even as a basis for a proof (see Mathematical induction[^]) As for morphisms, concrete examples can only help as counter examples, or to illustrate some behaviour. But morphisms can be very odd, and you can't really make a statement about them by looking just at examples - simply because you can never be sure whether you considered all relevant cases. What you have to do instead is analyze the facts you know about the morphisms in question, and, only based on these facts, consider the logical consequences. I'll give an example of a proof from a mathematical school contest I did 40 years ago. I still remember it because the statement to proof is so beautifully simple, and yet only a few 100 pupils in all of Germany were able to solve it correctly (i hope my translation skills don't let me down: Given is a bijective morphism of the euclidian plane, f : R^2 -> R^2 that projects any circle onto a circle. Prove that f will also project any straight line onto a straight line. A lot of pupils failed on this proof because of one mistake: they assumed that f would not only project circles on to circle, but also the center of the circle onto the center of the projected circle. However, the task does not give that information, and a proof based on that assumption is therefore wrong. The solution takes several steps: 1. consider the method: there are several method to do mathematical proofs, but the only one I could come up with that fits this task was Reductio ad absurdum - Wikipedia[^] : I will assume the opposite of the statement. Then I will disprove this assumption. 2. Carefully formulate your assumption: The opposite of the statement is that not all straight lines are projected onto straight lines, or that there is at least one case of a straight line that is not projected onto a st
-
I would have to look up some translations to be precise, but it looks like your question is much more general than something about specific types of morphism: First of all, while Mathemtatics is mostly about the art to describe and solve problems separate from concrete examples, examples are still often used to illustrate corner cases, to disprove a false assumption, and occasionally even as a basis for a proof (see Mathematical induction[^]) As for morphisms, concrete examples can only help as counter examples, or to illustrate some behaviour. But morphisms can be very odd, and you can't really make a statement about them by looking just at examples - simply because you can never be sure whether you considered all relevant cases. What you have to do instead is analyze the facts you know about the morphisms in question, and, only based on these facts, consider the logical consequences. I'll give an example of a proof from a mathematical school contest I did 40 years ago. I still remember it because the statement to proof is so beautifully simple, and yet only a few 100 pupils in all of Germany were able to solve it correctly (i hope my translation skills don't let me down: Given is a bijective morphism of the euclidian plane, f : R^2 -> R^2 that projects any circle onto a circle. Prove that f will also project any straight line onto a straight line. A lot of pupils failed on this proof because of one mistake: they assumed that f would not only project circles on to circle, but also the center of the circle onto the center of the projected circle. However, the task does not give that information, and a proof based on that assumption is therefore wrong. The solution takes several steps: 1. consider the method: there are several method to do mathematical proofs, but the only one I could come up with that fits this task was Reductio ad absurdum - Wikipedia[^] : I will assume the opposite of the statement. Then I will disprove this assumption. 2. Carefully formulate your assumption: The opposite of the statement is that not all straight lines are projected onto straight lines, or that there is at least one case of a straight line that is not projected onto a st
Very well written! The core of my question lies around this:
Stefan_Lang wrote:
What you have to do instead is analyze the facts you know about the morphisms in question, and, only based on these facts, consider the logical consequences.
Say we're trying to show a morphism f is monomorphic, then it has to satisfy:
f:X -> Y such that for all objects Z and all morphisms g1, g2: Z -> X, f o g1 = f o g2 => g1 = g2
We can base the equality of morphisms in general around the equality of functions:
Let X and Y be object classes and f:X -> Y and g:X -> Y be morphisms. We say that f and g are equal and write f=g if f(a)=g(a) for all a in X.
Regardless of the approach towards a proof we choose, there must be consideration given to the "a in X." This necessitates making the object classes concrete I believe (e.g. you can no longer have just an object class "cars" since now you need to consider the equivalence classes within "cars"; we have to consider mappings, not just domain and codomain). And I think this is where I start getting a little confused. If not considering a concrete category, we don't have enough information. Maybe it's just as simple as "showing properties like this requires a concrete category," but then I don't see as much benefit to the more abstract views of categories. I'll continue to read and think about it. I'm sure it'll click eventually. Thanks for the response! I do need to get better at proofs in general. It would make digesting some of these books a lot easier.