Examples for the Yoneda lemma
Hello, I am looking for (simple) instructive examples for the Yoneda lemma, showing how to get the "inner" structure of an object from its morphisms. I've been told how to get a graph G from its morphisms (from the one-vertex-graph V to G and the one-edge-graph E to G and the morphisms from V to E) and appreciated this example a lot. Are there others equally simple and enlightening? What I wonder is which morphisms are definitely needed. In the graph example it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted and generalized? Many thanks in advance Hans-Stricker [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Hans-Peter, In the case you mention, the category of graphs is a presheaf category (the category of all functors C^op-->Set, for a small category C. The graphs V and E you mention are precisely the representable functors, and the morphisms V to E are the morphisms between these. This is a general phenomenon: for any presheaf category [C^op,Set], you can recover an object X when you know all the morphisms to X from the representables, as well as how these morphisms behave when composed with morphisms between representables. The key property here is that the representables form a dense full subcategory of [C^op,Set], and there is a further generalization to that setting. For another concrete example, consider the category Ab of abelian groups. The elements of a group A can be identified with the morphisms Z-->A. Pairs of elements can be identified with morphisms Z^2-->A, and the group operation can then be recovered using the diagonal map Z-->Z^2. The other operations can be recovered similarly (you'd also want to use the trivial group in order to recover the unit element). If you also want to check the associative law, you'd want to use the object Z^3 as well. The connection between the previous two paragraphs is that the full subcategory of Ab consisting of Z, Z^2, Z^3, and 1 is dense. (Actually, you could just use Z and Z^2 for this.) A similar analysis can be done for the models of any Lawvere theory in place of Abelian groups. Steve Lack. On 15/01/10 11:24 AM, "Hans-Peter Stricker" <stricker@epublius.de> wrote:
Hello,
I am looking for (simple) instructive examples for the Yoneda lemma, showing how to get the "inner" structure of an object from its morphisms. I've been told how to get a graph G from its morphisms (from the one-vertex-graph V to G and the one-edge-graph E to G and the morphisms from V to E) and appreciated this example a lot. Are there others equally simple and enlightening?
What I wonder is which morphisms are definitely needed. In the graph example it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted and generalized?
Many thanks in advance
Hans-Stricker
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Lots of examples in http://boole.stanford.edu/pub/yon.pdf . It's an 18-page paper, yet already by page 2 there are six examples, none of them the usual graph example. In coming to grips with those examples it is *very* helpful to realize that every algebraic theory including the well-known ones having at least one constant or constant operation has a unary subtheory obtained by fixing all but one argument of every nonzeroary operation in the clone (which will be just projection if and only if all nonzeroary operations are projections). And while you don't need it on page 2, further on it is helpful to realize that for every algebraic theory, its models form a full subcategory of a presheaf category. In order to reach a broader audience the paper was written for an algebraic audience. If you're more familiar with category language than algebraic language a little adaptation will be needed. Let me know if translating back into category theory gives any trouble, this would be helpful feedback to have. Vaughan Pratt Hans-Peter Stricker wrote:
Hello,
I am looking for (simple) instructive examples for the Yoneda lemma, showing how to get the "inner" structure of an object from its morphisms. I've been told how to get a graph G from its morphisms (from the one-vertex-graph V to G and the one-edge-graph E to G and the morphisms from V to E) and appreciated this example a lot. Are there others equally simple and enlightening?
What I wonder is which morphisms are definitely needed. In the graph example it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted and generalized?
Many thanks in advance
Hans-Stricker
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The simplest example I can think of is posets. If you represent a poset as a category (i.e. a category with at most one arrow from A->B such that A->B and B->A implies A=B), then an object A is completely determined by the set of arrows going in to it. In this context, the Yoneda embedding is the familiar result that any poset P embeds fully and faithfully in the powerset of P, ordered by subset inclusion. Aleks On Fri, Jan 15, 2010 at 12:24 AM, Hans-Peter Stricker <stricker@epublius.de> wrote:
Hello,
I am looking for (simple) instructive examples for the Yoneda lemma, showing how to get the "inner" structure of an object from its morphisms. I've been told how to get a graph G from its morphisms (from the one-vertex-graph V to G and the one-edge-graph E to G and the morphisms from V to E) and appreciated this example a lot. Are there others equally simple and enlightening?
What I wonder is which morphisms are definitely needed. In the graph example it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted and generalized?
Many thanks in advance
Hans-Stricker
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hello Aleks, I am not quite what to think of the poset of unlabeled graphs without isolated vertices with the relation of embeddability: I have the feeling, that such a graph is NOT completely determined by its set of in-arrows (see http://epublius.de/Fragment_of_the_category_of_unlabeled_graphs_without_isol... to see what I mean, e.g. vertices 3 and 4 or vertices 7,8,9). Do I miss something? Best Hans-Peter ----- Original Message ----- From: "Aleks Kissinger" <aleks0@gmail.com> To: "Hans-Peter Stricker" <stricker@epublius.de> Cc: <categories@mta.ca> Sent: Friday, January 15, 2010 12:07 PM Subject: Re: categories: Examples for the Yoneda lemma
The simplest example I can think of is posets. If you represent a poset as a category (i.e. a category with at most one arrow from A->B such that A->B and B->A implies A=B), then an object A is completely determined by the set of arrows going in to it.
In this context, the Yoneda embedding is the familiar result that any poset P embeds fully and faithfully in the powerset of P, ordered by subset inclusion.
Aleks
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Aleks Kissinger -
Hans-Peter Stricker -
Steve Lack -
Vaughan Pratt