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Atrium > Mathematics, Physics & Philosophy: Question: expressing linear independence in category theory?

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- CommentRowNumber1.
- CommentAuthorIan_Durham
- CommentTimeApr 4th 2010
- (edited Apr 4th 2010)
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Author: Ian_Durham
Format: HtmlSuppose there exists a set of matrices, Ak, that are linearly independent (that is, each matrix is linearly independent of all the other matrices in the set). How would one represent such a set in category theoretic terms? (This relates to the particularly sticky problem of how to categorically describe [[extremal quantum channels]].)
Suppose there exists a set of matrices, Ak, that are linearly independent (that is, each matrix is linearly independent of all the other matrices in the set). How would one represent such a set in category theoretic terms? (This relates to the particularly sticky problem of how to categorically describe extremal quantum channels.)
- CommentRowNumber2.
- CommentAuthorHarry Gindi
- CommentTimeApr 4th 2010
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Author: Harry Gindi
Format: MarkdownYou may as well just title your posts "Vague Title"!

You may as well just title your posts "Vague Title"!
- CommentRowNumber3.
- CommentAuthorIan_Durham
- CommentTimeApr 4th 2010
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Author: Ian_Durham
Format: HtmlHarry: I actually realized that after I had posted it, but then couldn't find a way to change the title of the post.
Harry: I actually realized that after I had posted it, but then couldn't find a way to change the title of the post.
- CommentRowNumber4.
- CommentAuthorHarry Gindi
- CommentTimeApr 4th 2010
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Author: Harry Gindi
Format: MarkdownClick "edit" on the first post, then change title at the top.

Click "edit" on the first post, then change title at the top.
- CommentRowNumber5.
- CommentAuthorIan_Durham
- CommentTimeApr 4th 2010
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Author: Ian_Durham
Format: HtmlHarry: thanks. Didn't think that would allow me to change the title, but obviously it did.
Harry: thanks. Didn't think that would allow me to change the title, but obviously it did.
- CommentRowNumber6.
- CommentAuthorEric
- CommentTimeApr 4th 2010
- (edited Apr 4th 2010)
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Author: Eric
Format: MarkdownI don't know either, but here are some thoughts to maybe push things forward a small step. There is a category [[Vect]] whose objects are vector spaces and morphisms are linear maps. There is a category [[Set]] whose objects are sets and morphisms are functions. There is a [[forgetful functor]] <latex>U:Vect\to Set</latex> which takes a vector space <latex>V</latex> and maps it to the underlying "set" of vectors. Whenever you have a forgetful functor laying around, it is usually interesting to look at its [[adjoint functor]] which goes the other way <latex>F:Set\to Vect</latex> This takes a "set" <latex>A</latex> and gives a vector space. In physics/engineering, you'd say the "vector space spanned by <latex>A</latex>". So the beginning of your question could be rephrased to something like "Say we have a set <latex>A</latex> of matrices <latex>A_k</latex> and form the free vector space <latex>F(A)</latex>." I'm not sure where to go from here other than maybe to point out that the determinant <latex>det</latex> is a functor. Ok. Hopefully an expert can step in now :)

I don't know either, but here are some thoughts to maybe push things forward a small step.

There is a category Vect whose objects are vector spaces and morphisms are linear maps.

There is a category Set whose objects are sets and morphisms are functions.

There is a forgetful functor
$U:Vect\to Set$
which takes a vector space $V$ and maps it to the underlying "set" of vectors.

Whenever you have a forgetful functor laying around, it is usually interesting to look at its adjoint functor which goes the other way
$F:Set\to Vect$
This takes a "set" $A$ and gives a vector space. In physics/engineering, you'd say the "vector space spanned by $A$ ".

So the beginning of your question could be rephrased to something like "Say we have a set $A$ of matrices $A_k$ and form the free vector space $F(A)$ ."

I'm not sure where to go from here other than maybe to point out that the determinant $det$ is a functor.

Ok. Hopefully an expert can step in now :)
- CommentRowNumber7.
- CommentAuthorzskoda
- CommentTimeApr 4th 2010
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Author: zskoda
Format: MarkdownA general nonsense formalism, not in the sense of category theory, to bases and linear independence, has been axiomatized by a Bourbaki member Whitney, under the name "matroid". Do we have an entry [[matroid]] ?

A general nonsense formalism, not in the sense of category theory, to bases and linear independence, has been axiomatized by a Bourbaki member Whitney, under the name "matroid". Do we have an entry matroid ?
- CommentRowNumber8.
- CommentAuthorIan_Durham
- CommentTimeApr 5th 2010
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Author: Ian_Durham
Format: HtmlApparently we don't have an entry on matroid. I've heard of matroids though. I'll have to look them up. I like where Eric is going with this, though. Perhaps some sort of composition of the forgetful functor described above and its adjoint could be used to capture the linear independence condition I'm looking for.
Apparently we don't have an entry on matroid. I've heard of matroids though. I'll have to look them up.

I like where Eric is going with this, though. Perhaps some sort of composition of the forgetful functor described above and its adjoint could be used to capture the linear independence condition I'm looking for.
- CommentRowNumber9.
- CommentAuthorTodd_Trimble
- CommentTimeApr 5th 2010
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Author: Todd_Trimble
Format: MarkdownWell, just to pick up where Eric left off: suppose <latex>I</latex> is an index set for a collection of vectors in a vector space <latex>V</latex> (for example, <latex>V</latex> could be a vector space of matrices). In other words, suppose we have a function <latex>A: I \to V</latex> which takes <latex>k \in I</latex> to <latex>A_k \in V</latex>. In other words, suppose we have a function <latex>A: I \to U(V)</latex>, where <latex>U(V)</latex> is the underlying set of <latex>V</latex> (<latex>U: Vect \to Set</latex> being the functor which forgets vector space structure). Then, by the free-forgetful adjunction to which Eric refers, the function <latex>A: I \to U(V)</latex> corresponds uniquely to a linear map <latex>\hat{A}: F(I) \to V</latex>. The collection <latex>\{A_k\}</latex> is linearly independent if <latex>\hat{A}</latex> is an injection (a monomorphism); equivalently, if the kernel of <latex>\hat{A}</latex> is trivial.

Well, just to pick up where Eric left off: suppose $I$ is an index set for a collection of vectors in a vector space $V$ (for example, $V$ could be a vector space of matrices). In other words, suppose we have a function $A: I \to V$ which takes $k \in I$ to $A_k \in V$ . In other words, suppose we have a function $A: I \to U(V)$ , where $U(V)$ is the underlying set of $V$ ( $U: Vect \to Set$ being the functor which forgets vector space structure).

Then, by the free-forgetful adjunction to which Eric refers, the function $A: I \to U(V)$ corresponds uniquely to a linear map $\hat{A}: F(I) \to V$ .

The collection $\{A_k\}$ is linearly independent if $\hat{A}$ is an injection (a monomorphism); equivalently, if the kernel of $\hat{A}$ is trivial.
- CommentRowNumber10.
- CommentAuthorIan_Durham
- CommentTimeApr 5th 2010
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Author: Ian_Durham
Format: HtmlSo presumably extending this to the collection <latex>\{A^{\dag}_{k}A_{k}\}</latex> is relatively trivial as is, perhaps, extending this to <latex>\{A^{\dagger}_{k}A_{k} \oplus A_{k}A^{\dagger}\}</latex>. The question is how to check if these conditions hold over a tensor product (functor).
So presumably extending this to the collection is relatively trivial as is, perhaps, extending this to . The question is how to check if these conditions hold over a tensor product (functor).
- CommentRowNumber11.
- CommentAuthorTodd_Trimble
- CommentTimeApr 5th 2010
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Author: Todd_Trimble
Format: MarkdownCould you make this question about the tensor product more explicit?

Could you make this question about the tensor product more explicit?
- CommentRowNumber12.
- CommentAuthorIan_Durham
- CommentTimeApr 6th 2010
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Author: Ian_Durham
Format: HtmlSure. So, suppose we show that the set of matrices (or, more appropriately, operators with matrix representation) <latex>\{A^{\dag}_{k}A_{l}\}_{k,l\ldots N}</latex> is a linearly independent set. There are a few specific examples from physics where the set <latex>\left\{\left(A^{\dag}_{k1}\otimes\cdots\otimes A^{\dag}_{kN}\right)\left(A_{l1}\otimes\cdots\otimes A_{lN}\right)\right\}_{k1,l1,\ldots kN,lN}</latex> is <em>not</em> linearly independent. The question is if there is a way to check in going from the first to the second, if the linear independence condition holds.
Sure. So, suppose we show that the set of matrices (or, more appropriately, operators with matrix representation) is a linearly independent set. There are a few specific examples from physics where the set is not linearly independent. The question is if there is a way to check in going from the first to the second, if the linear independence condition holds.

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Atrium > Mathematics, Physics & Philosophy: Question: expressing linear independence in category theory?