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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 25th 2019

    Minor edit to trigger a discussion thread.

    diff, v14, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 25th 2019

    Previous discussion, here and here.

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeMar 3rd 2020

    whoever before has wondered about the elusive integral notation for the coend, your time has come :)

    Consider this example: let CC be the category of finite dimensional vector spaces over a field kk. Let F:C×C opCF : C \times C^{op} \rightarrow C be the functor sending (V,W)(V, W) to VW *V \otimes W^*. Then VCF(V,V)k\int^{V \in C} F(V, V) \cong k. The structure maps ε V\epsilon_V in the coend are ev:VV *k\text{ev} : V \otimes V^* \rightarrow k sending vfv \otimes f to f(v)f(v).

    In this case, and others, the recipient of the integration pairing is canonically a coend of Hom(W,V)V *W\text{Hom}(W, V) \cong V^* \otimes W. What I am (hesitantly) suggesting is that the coend

    VCV *V\int^{V \in C} V^* \otimes V

    and its structure maps

    ε X:V *Vk \epsilon_X : V^* \otimes V \rightarrow k

    are the universal construction by which the intensives and the extensives are integrated against each other.

    Note: the coend CHom(W,V)\int^C \text{Hom}(W, V) is far more general than the integration pairing, but seems to match it in many cases.

    Little conjecture: Let DD be the category of Banach spaces with short maps as morphisms. Let F:D×D opDF : D \times D^{op} \rightarrow D be the functor sending (V,W)(V, W) to V^W *V \hat{\otimes} W^*. Then VDF(V,V)\int^{V \in D} F(V, V) \cong \mathbb{R}.The structure maps ε V\epsilon_V are ev:V^V *k\text{ev} : V \hat{\otimes} V^* \rightarrow k sending vfv \otimes f to f(v)f(v). Let CH be the category of compact hausdorff spaces and let XX be an object in CH. Let V=[X,] CHV = [X, \mathbb{R}]_{\text{CH}}, an object in DD. An element in V *V^* is a choice of integral (a choice of which extensive property to integrate against), and for ϕV *\phi \in V^* and fVf \in V,

    ϕ(v)=ev(vϕ)=:vdϕ\phi(v) = \text{ev} (v \otimes \phi) =: \int v d \phi

    Your thoughts?

    • CommentRowNumber4.
    • CommentAuthorDean
    • CommentTimeMar 3rd 2020

    (P.S. I wrote the last post, but I wasn’t signed in.)

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMar 3rd 2020
    Oh, and in the above little conjecture, the dual spaces are spaces of bounded maps.
    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 3rd 2020

    Loregian’s book on coends opens with this kind of comparison.

    Hmm, with the end as ’subspace of invariants for the action’ and coend as ’the space of orbits of said action’ is a HoTT rendition possible? I guess a starting point is:

    In complete analogy to how limits are right adjoint functors to the diagonal functor, ends are right adjoint functors to the hom functor.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 3rd 2020
    • (edited Mar 3rd 2020)

    I see there’s section 2.2.2 of Combinatorial species and labelled structures on ’Coends in HoTT’. [Right, I see the Haskell community write as homotopy (co)limits: exists x. p x x, forall x. p x x.]

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeMar 4th 2020

    Proposition: Let VV be a monoidal closed category with tensor V\otimes_V and internal hom [X,Y] V[X, Y]_V. Suppose that the Kan extension Lan Id V(Id V)\text{Lan}_{\text{Id}_V} (\text{Id}_V) exists and is pointwise. Then

    XId V(X)Lan Id V(Id V)(X) YV[Y,X] V VYX \cong \text{Id}_V (X) \cong \text{Lan}_{\text{Id}_V} (\text{Id}_V) (X) \cong \int^{Y \in V} [Y, X]_V \otimes_V Y

    Let Ban 1\text{Ban}_1 be the category of Banach spaces and short maps. Ban 1\text{Ban}_1 has an internal Hom consisting of bounded maps (Write [,][-, -] for external Hom and Hom(,)\text{Hom}(-, -) for internal Hom). Internal Hom has a left adjoint, projective tensor product.

    Theorem: Lan Id Ban 1(Id Ban 1)\text{Lan}_{\text{Id}_{\text{Ban}_1}}(\text{Id}_{\text{Ban}_1}) exists and is pointwise, so that

    Id D()Lan Id D(Id D) VD[V,] DV\mathbb{R} \cong \text{Id}_D (\mathbb{R}) \cong \text{Lan}_{\text{Id}_D} (\text{Id}_D) \cong \int^{V \in D} [V, \mathbb{R}] \otimes_D V

    where D\otimes_D is projective tensor product.

    • CommentRowNumber9.
    • CommentAuthorGuest
    • CommentTimeMar 4th 2020
    That should be $\int^{V \in D} \text{Hom}(V, \mathbb{R}) \otimes_D V$.

    I will add this and its proof -- is that ok?
  1. I have added the coend-integral comparison to section 3 of the intensive/extensive property page.

    This includes the theorem mentioned in (8) in our nform discussion here.

    I wrote it in terms of Lawvere metrics instead of metrics.

    diff, v15, current