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Math Help - Help on Category Theory

  1. #1
    Super Member redsoxfan325's Avatar
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    Help on Category Theory

    I'm taking an advanced abstract algebra course this year, and the professor decided to give us some homework on category theory without actually teaching it to us. The next time we meet (Wednesday) is when the homework is due.

    I think I understand what a morphism is: A set of all maps from one object to another (in a given category). I understand the definition of initial and final objects, but I can't think of any examples of them OR how to be able to find them given a category. For example, one question on the homework is "Indicate the initial and final objects of the category Vect(K) of finite-dimensional vector spaces over the field K."

    Two other related concepts I don't understand are that of a direct sum and canonical embeddings. We need to show that in the category Vect(K), direct sums exists for all pairs of objects.

    Please help! I am pulling my hair out!
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    Quote Originally Posted by redsoxfan325 View Post
    I'm taking an advanced abstract algebra course this year, and the professor decided to give us some homework on category theory without actually teaching it to us. The next time we meet (Wednesday) is when the homework is due.

    I think I understand what a morphism is: A set of all maps from one object to another (in a given category). I understand the definition of initial and final objects, but I can't think of any examples of them OR how to be able to find them given a category. For example, one question on the homework is "Indicate the initial and final objects of the category Vect(K) of finite-dimensional vector spaces over the field K."

    Two other related concepts I don't understand are that of a direct sum and canonical embeddings. We need to show that in the category Vect(K), direct sums exists for all pairs of objects.

    Please help! I am pulling my hair out!
    A morphism is an arrow from one object to another, not the whole set (assuming your category is locally small).

    Remember that an initial object is an object A in your category \mathcal{C} such that for every object B in \mathcal{C} there is a unique morphism f:A \longrightarrow B. If you're in \mathcal{VECT(K)} then \{ 0 \} clearly satisfies the definition, since a morphism in \mathcal{VECT(K)} is a linear transformation. Similarly you can show that \{ 0 \} is the terminal object.

    For the second part I haven't reached those topics so I really can't say.
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  3. #3
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    the product of two objects A,B in a category \mathcal{C} is an object C and morphisms p_1: C \longrightarrow A, \ p_2: C \longrightarrow B such that for every object D and morphisms f: D \longrightarrow A

    and g: D \longrightarrow B, there exists a unique morphism h: D \longrightarrow C such that our diagram commutes, i.e. p_1h=f, \ p_2h=g.

    the reason that i showed the morphisms by p_1,p_2 is that they really behave like "projections". anyway, it's not hard to prove that the direct sum (or product) is unique if it

    exists. for the category \mathcal{VECT}(K) define C=A \oplus B, where \oplus is the ordinary direct product of vector spaces. also define p_1(a,b)=a, \ p_2(a,b)=b, for all a \in A, \ b \in B.

    now for a (finite dimensional) K-vector space D and linear transformations f: D \longrightarrow A, \ g: D \longrightarrow B define h: D \longrightarrow A \oplus B by h(d)=(f(d),g(d)). clearly h is a linear

    transformation because f,g are and p_1h=f, \ p_2h=g. if u: D \longrightarrow A \oplus B is another such linear transformation, then assuming that u(d)=(\alpha(d), \beta(d)), we will have

    f(d)=p_1u(d)=\alpha(d), \ g(d)=p_2u(d)=\beta(d). thus u = h. so h is unique and we're done.


    Edit: Ok, i just noticed that your question was about "direct sum", which i think you mean "coproduct". this is just the "dual" of the product. so you just reverse the direction

    of all arrows to get the definition of coproduct. for the category K-vector spaces (more generally R-modules) product and coproduct of a finitely many of objects are the same.
    Last edited by NonCommAlg; September 14th 2009 at 05:29 AM.
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    Quote Originally Posted by NonCommAlg View Post
    the product of two objects A,B in a category \mathcal{C} is an object C and morphisms p_1: C \longrightarrow A, \ p_2: C \longrightarrow B such that for every object D and morphisms f: D \longrightarrow A

    and g: D \longrightarrow B, there exists a unique morphism h: D \longrightarrow C such that our diagram commutes, i.e. p_1h=f, \ p_2h=g.

    the reason that i showed the morphisms by p_1,p_2 is that they really behave like "projections". anyway, it's not hard to prove that the direct sum (or product) is unique if it

    exists. for the category \mathcal{VECT}(K) define C=A \oplus B, where \oplus is the ordinary direct product of vector spaces. also define p_1(a,b)=a, \ p_2(a,b)=b, for all a \in A, \ b \in B.

    now for a (finite dimensional) K-vector space D and linear transformations f: D \longrightarrow A, \ g: D \longrightarrow B define h: D \longrightarrow A \oplus B by h(d)=(f(d),g(d)). clearly h is a linear

    transformation because f,g are and p_1h=f, \ p_2h=g. if u: D \longrightarrow A \oplus B is another such linear transformation, then assuming that u(d)=(\alpha(d), \beta(d)), we will have

    f(d)=p_1u(d)=\alpha(d), \ g(d)=p_2u(d)=\beta(d). thus u = h. so h is unique and we're done.


    Edit: Ok, i just noticed that your question was about "direct sum", which i think you mean "coproduct". this is just the "dual" of the product. so you just reverse the direction

    of all arrows to get the definition of coproduct. for the category K-vector spaces (more generally R-modules) product and coproduct of a finitely many of objects are the same.
    In wiki,
    ".. in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only "finitely" many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors). "

    If we drop the "finitely many" condition for the direct product of abelian groups, why it is not the coproduct of the category of abelian groups?
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  5. #5
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    Quote Originally Posted by aliceinwonderland View Post

    If we drop the "finitely many" condition for the direct product of abelian groups, why it is not the coproduct of the category of abelian groups?
    before i answer your question let me give the definition of product and coproduct for the most general case, i.e. for a "family" \{A_j \}_{j \in I} of objects in a category \mathcal{C}:


    1) the product is an objcet B and morphims p_j: B \longrightarrow A_j, \ j \in I, such that for every object D and morphisms f_j : D \longrightarrow A_j, \ j \in I, there exists a unique morphism

    g: D \longrightarrow B such that p_jg=f_j, \ j \in I.


    2) the coproduct is an object C and morphisms \iota_j: A_j : \longrightarrow C, \ j \in I, such that for every object D and morphisms f_j : A_j \longrightarrow D, \ j\in I, there exists a unique morphism

    g: C \longrightarrow D such that g\iota_j=f_j, \ j \in I.


    from now on i'll assume that \mathcal{C} is the category of all vector spaces over a field K [or the category of abelian groups or in general the category of R modules, where R is a

    (commutative) ring]. then, exactly as i showed in my previous post, \prod_{j \in I} A_j will be the product of A_j, \ j \in I. clearly if \mathcal{C} is the category of finite dimensional vector spaces over

    a field K, then the index set I has to be finite because otherwise the vector space \prod_{j \in I} A_j will not be in \mathcal{C} anymore.

    for the coproduct let C=\oplus_{j \in I} A_j and define \iota_j : A_j \longrightarrow C, \ j \in I, to be the natural injection. let p_j: C \longrightarrow A_j, \ j \in I, be the natural projection. now for any D \in \mathcal{C}, and

    morphisms f_j : A_j \longrightarrow D define g: C \longrightarrow D by g(a) = \sum_{i \in I} f_ip_i(a). this definition is well-defined only if only we have finitely many term in the sum, which explains why we've

    considered \oplus instead of \prod. (this also answers aliceinwonderland's question!) now for any j \in I we have g \iota_j(a_j)=\sum_{i \in I} f_ip_i(\iota_j(a_j))=f_jp_j(\iota_j(a_j))=f_j(a_j)  . thus g \iota_j = f_j

    for all j \in I. so, in order to prove that C=\oplus_{j \in I} A_j is the coproduct of A_j, \ j \in I, we only need to prove the uniqueness of g, which is easy.

    again, note that if \mathcal{C} is the category of finite dimensional vector spaces over some field K, then the index set I will have to be finite.
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    Quote Originally Posted by NonCommAlg View Post
    before i answer your question let me give the definition of product and coproduct for the most general case, i.e. for a "family" \{A_j \}_{j \in I} of objects in a category \mathcal{C}:


    1) the product is an objcet B and morphims p_j: B \longrightarrow A_j, \ j \in I, such that for every object D and morphisms f_j : D \longrightarrow A_j, \ j \in I, there exists a unique morphism

    g: D \longrightarrow B such that p_jg=f_j, \ j \in I.


    2) the coproduct is an object C and morphisms \iota_j: A_j : \longrightarrow C, \ j \in I, such that for every object D and morphisms f_j : A_j \longrightarrow D, \ j\in I, there exists a unique morphism

    g: C \longrightarrow D such that g\iota_j=f_j, \ j \in I.


    from now on i'll assume that \mathcal{C} is the category of all vector spaces over a field K [or the category of abelian groups or in general the category of R modules, where R is a

    (commutative) ring]. then, exactly as i showed in my previous post, \prod_{j \in I} A_j will be the product of A_j, \ j \in I. clearly if \mathcal{C} is the category of finite dimensional vector spaces over

    a field K, then the index set I has to be finite because otherwise the vector space \prod_{j \in I} A_j will not be in \mathcal{C} anymore.

    for the coproduct let C=\oplus_{j \in I} A_j and define \iota_j : A_j \longrightarrow C, \ j \in I, to be the natural injection. let p_j: C \longrightarrow A_j, \ j \in I, be the natural projection. now for any D \in \mathcal{C}, and

    morphisms f_j : A_j \longrightarrow D define g: C \longrightarrow D by g(a) = \sum_{i \in I} f_ip_i(a). this definition is well-defined only if only we have finitely many term in the sum, which explains why we've

    considered \oplus instead of \prod. (this also answers aliceinwonderland's question!) now for any j \in I we have g \iota_j(a_j)=\sum_{i \in I} f_ip_i(\iota_j(a_j))=f_jp_j(\iota_j(a_j))=f_j(a_j)  . thus g \iota_j = f_j

    for all j \in I. so, in order to prove that C=\oplus_{j \in I} A_j is the coproduct of A_j, \ j \in I, we only need to prove the uniqueness of g, which is easy.

    again, note that if \mathcal{C} is the category of finite dimensional vector spaces over some field K, then the index set I will have to be finite.
    I understand that if the category C is the category of finitely generated abelian groups, then both direct product and direct sum can be a coproduct in C.

    If Ab is the category of abelian groups in a general sense, why the direct product is not a coproduct in Ab? Any example?

    Thanks.
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    Quote Originally Posted by aliceinwonderland View Post

    I understand that if the category C is the category of finitely generated abelian groups, then both direct product and direct sum can be a coproduct in C.
    yes, because for the category of f.g. abelian groups, the product (coproduct) is guaranteed to exist for a finite set of the objects of that category and obviously \bigoplus_{j=1}^n A_j \cong \prod_{j=1}^n A_j.


    If Ab is the category of abelian groups in a general sense, why the direct product is not a coproduct in Ab? Any example?

    Thanks.
    if the coproduct (or product) of a family of objects in a category exists, it'll have to be unique. for \text{Ab}, as we showed, the direct sum is the coproduct. so your question can be rephrased

    in this way: can we find an example of a family of abelian groups A_j, \ j \in I, such that \bigoplus_{j \in I} A_j \ncong \prod_{j \in I} A_j ? the answer is yes, there are millions of such examples. here's an obvious one:

    let A_j = \mathbb{Z}/j \mathbb{Z}, \ j \in \mathbb{N}. then every element of \bigoplus_{j=1}^{\infty} A_j has finite order but this is clearly not true in \prod_{j=1}^{\infty} A_j.
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  8. #8
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    Quote Originally Posted by NonCommAlg View Post
    the product of two objects A,B in a category \mathcal{C} is an object C and morphisms p_1: C \longrightarrow A, \ p_2: C \longrightarrow B such that for every object D and morphisms f: D \longrightarrow A

    and g: D \longrightarrow B, there exists a unique morphism h: D \longrightarrow C such that our diagram commutes, i.e. p_1h=f, \ p_2h=g.

    the reason that i showed the morphisms by p_1,p_2 is that they really behave like "projections". anyway, it's not hard to prove that the direct sum (or product) is unique if it

    exists. for the category \mathcal{VECT}(K) define C=A \oplus B, where \oplus is the ordinary direct product of vector spaces. also define p_1(a,b)=a, \ p_2(a,b)=b, for all a \in A, \ b \in B.

    [color="blue"]now for a (finite dimensional) K-vector space D and linear transformations f: D \longrightarrow A, \ g: D \longrightarrow B define h: D \longrightarrow A \oplus B by h(d)=(f(d),g(d)). clearly h is a linear

    transformation because f,g are and p_1h=f, \ p_2h=g. if u: D \longrightarrow A \oplus B is another such linear transformation, then assuming that u(d)=(\alpha(d), \beta(d)), we will have

    f(d)=p_1u(d)=\alpha(d), \ g(d)=p_2u(d)=\beta(d). thus u = h. so h is unique and we're done.


    Edit: Ok, i just noticed that your question was about "direct sum", which i think you mean "coproduct". this is just the "dual" of the product. so you just reverse the direction

    of all arrows to get the definition of coproduct. for the category K-vector spaces (more generally R-modules) product and coproduct of a finitely many of objects are the same.
    OK, I've spent the last day trying to wrap my head around this. You say that for direct sum I should just reverse the arrows, but the functions don't really make sense any more, do they? For instance, p_1:C\longrightarrow A. You say define p_1(a,b)=a, which is fine, but if I want p_1: A\longrightarrow C, what is p_1 now?

    If I have two vector spaces U and V what would morphisms i:U\longrightarrow (U\oplus V) and j:V\longrightarrow (U\oplus V) be? Or what about \phi: (U\oplus V)\longrightarrow W, where W is another vector space? I am really struggling with this.
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  9. #9
    Super Member redsoxfan325's Avatar
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    I've attached a picture of the problem below.
    Attached Thumbnails Attached Thumbnails Help on Category Theory-problem-4.jpg  
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    Quote Originally Posted by redsoxfan325 View Post
    OK, I've spent the last day trying to wrap my head around this. You say that for direct sum I should just reverse the arrows, but the functions don't really make sense any more, do they? For instance, p_1:C\longrightarrow A. You say define p_1(a,b)=a, which is fine, but if I want p_1: A\longrightarrow C, what is p_1 now?
    you're right that just reversing the direction of the arrows is not enough. as i explained in # 5 in this thread, the morphisms this time will be the natural injections \iota_1: A_1 \longrightarrow C and \iota_2 : A_2 \longrightarrow C,

    where C=A_1 \oplus A_2 and \iota_1, \ \iota_2 are defined by \iota_1(a_1)=(a_1,0), \ \iota_2(a_2)=(0,a_2), for all a_1 \in A_1, \ a_2 \in A_2. now if A_1,A_2 are, say (finite dimensional) vector spaces over some field K, then C will

    also be a (finite dimensional) vector space over K, i.e. C is an object in our category. also it's clear that \iota_1, \iota_2 are K-linear transformations, i.e. they are morphisms in our category. to prove that

    C is a coproduct of A_1,A_2, we need to prove that for any (finite dimensional) K-vector space D and any linear transformations f_j : A_j \longrightarrow D, \ j=1,2, there exists a unique linear transformation
    g: C \longrightarrow D with this property that g \iota_j = f_j, \ j=1,2. here's how to prove it:

    existence of g: define g: C \longrightarrow D by g(a)=f_1p_1(a) + f_2p_2(a), where p_j : C \longrightarrow A_j, \ j=1,2, are the natural projections. it's clear that g is a linear transformation because f_j,p_j are. so we

    now need to show that g \iota_j = f_j, \ j=1,2. so suppose a_1 \in A_1. then g \iota_1 (a_1)=g(a_1,0)=f_1p_1(a_1,0) + f_2p_2(a_1,0)=f_1(a_1) + f_2(0)=f_1(a_1). similarly we see that g \iota_2 = f_2.

    uniqueness of g: suppose h: C \longrightarrow D is any morphism with this property that h \iota_j = f_j, \ j=1,2. we need to show that h=g: let a=(a_1,a_2) \in C. then:

    h(a)=h(a_1,0) + h(0,a_2)=h \iota_1(a_1) + h \iota_2(a_2)=f_1(a_1)+f_2(a_2). but we have a_1=p_1(a), \ a_2 = p_2(a) and thus h(a)=f_1p_1(a) + f_2p_2(a) = g(a). \ \Box


    Quote Originally Posted by redsoxfan325 View Post

    I've attached a picture of the problem below.
    this definition of coproduct (direct sum) is exactly the one that i gave you but in a more categorical language. the definition that i gave you is more standard.
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    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    you're right that just reversing the direction of the arrows is not enough. as i explained in # 5 in this thread, the morphisms this time will be the natural injections \iota_1: A_1 \longrightarrow C and \iota_2 : A_2 \longrightarrow C,

    where C=A_1 \oplus A_2 and \iota_1, \ \iota_2 are defined by \iota_1(a_1)=(a_1,0), \ \iota_2(a_2)=(0,a_2), for all a_1 \in A_1, \ a_2 \in A_2. now if A_1,A_2 are, say (finite dimensional) vector spaces over some field K, then C will

    also be a (finite dimensional) vector space over K, i.e. C is an object in our category. also it's clear that \iota_1, \iota_2 are K-linear transformations, i.e. they are morphisms in our category. to prove that

    C is a coproduct of A_1,A_2, we need to prove that for any (finite dimensional) K-vector space D and any linear transformations f_j : A_j \longrightarrow D, \ j=1,2, there exists a unique linear transformation
    g: C \longrightarrow D with this property that g \iota_j = f_j, \ j=1,2. here's how to prove it:

    existence of g: define g: C \longrightarrow D by g(a)=f_1p_1(a) + f_2p_2(a), where p_j : C \longrightarrow A_j, \ j=1,2, are the natural projections. it's clear that g is a linear transformation because f_j,p_j are. so we

    now need to show that g \iota_j = f_j, \ j=1,2. so suppose a_1 \in A_1. then g \iota_1 (a_1)=g(a_1,0)=f_1p_1(a_1,0) + f_2p_2(a_1,0)=f_1(a_1) + f_2(0)=f_1(a_1). similarly we see that g \iota_2 = f_2.

    uniqueness of g: suppose h: C \longrightarrow D is any morphism with this property that h \iota_j = f_j, \ j=1,2. we need to show that h=g: let a=(a_1,a_2) \in C. then:

    h(a)=h(a_1,0) + h(0,a_2)=h \iota_1(a_1) + h \iota_2(a_2)=f_1(a_1)+f_2(a_2). but we have a_1=p_1(a), \ a_2 = p_2(a) and thus h(a)=f_1p_1(a) + f_2p_2(a) = g(a). \ \Box




    this definition of coproduct (direct sum) is exactly the one that i gave you but in a more categorical language. the definition that i gave you is more standard.
    Thank you very much.
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    Basic category theory help

    Dear 'aliceinwonderland',

    I'm sorry to bother you. I'm really desperate for help with some category theory questions. Would you mind if I emailed you with details?

    Thanks.

    Joe
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    Quote Originally Posted by ctnovice View Post
    Dear 'aliceinwonderland',

    I'm sorry to bother you. I'm really desperate for help with some category theory questions. Would you mind if I emailed you with details?

    Thanks.

    Joe
    Well, I am not an expert of category theory.
    Why don't you post your question here? Some people (including me) might be able to help you. If you question includes lots of arrows and diagrams, just attach a file to your post.
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    ok sure, thanks for the advice.
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    Quote Originally Posted by ctnovice View Post
    Could anyone please help with a basic category question I'm struggling with. I'm quite new to this world.

    Please see the attached document for the problem - "joe_question_a.doc"

    Thanks in advance.

    Joe.
    Let J be an index category and Sets for the category of sets; define a functor F:J-->Sets that maps every arrow of J to an inclusion. We see that F is a nested sequence of sets for your question such that (F_1 = X_1) \subset (F_2 = X_2) \subset ....
    Colimit of (a) is the union U of all sets X_n with cocone given by inclusion maps \phi_n : (X_n) \rightarrow U, where \phi_n^{-1}=e_n^{-1} \cdot e_{n+1}^{-1} \cdots

    For (b), the infinite product of X_n, \prod_j{ X_j}, with cone given by projection maps q_j : \prod_j{X_j} \rightarrow X_j, where q_n^{-1}=p_n^{-1} \cdot p_{n+1}^{-1}, \cdots

    To show that a universal property for (a), given cocone (U, \phi) and another cocone (V, \psi), there exists a unique morphism f:V \rightarrow U such that f\psi_j = \phi_j for all j in J.

    To show that a universal property for (b), given cone (\prod_j{ X_j}, q) and another cone (C, y), there exists a unique morphism f:\prod_j{ X_j} \rightarrow C such that y_jf = q_j for all j in J.

    Note: I don't guarantee that the above solution of mine is error-free. Compare your own solution and let me know if you find any error.
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