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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeOct 12th 2012

    added to closed monoidal category a proof that the pointwise tensor product on a functor category with complete codomain is closed.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2014

    Discussion elsewhere suggested that readers found the bi-closed aspect at closed monoidal category not clear enough, or not highlighted properly to be recognizable.

    Therefore I have now edited slightly, trying to make the point clearer. Also added a few more hyperlinks to the existing text.

    And added pointers to Lambek’s original articles to the References-section, together with a pointer to the history of the subject over at linear type theory.

    • CommentRowNumber3.
    • CommentAuthorColin Tan
    • CommentTimeJan 5th 2016

    As a symmetric closed monoidal category, a cartesian closed category (C,×,1)(C, \times, 1) has the properties that (1) the unit 11 is the terminal object and (2) the tensor product ×\times distributes over finite coproducts.

    Is there a name for symmetric closed monoidal categories that satisfy properties (1) and (2)?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 5th 2016
    • (edited Jan 5th 2016)

    (2) is redundant since in a symmetric monoidal closed category, it is automatic that the tensor product distributes over all colimits.

    As for (1): is it true that I’ve heard “affine monoidal”? (Terminology based on the example of affine spaces = vector spaces but forgetting the origin.) I’ll have to check later.

    Edit: Oh, the nLab calls it a semicartesian monoidal category, so you could call your thing a semicartesian closed category I guess.

    • CommentRowNumber5.
    • CommentAuthorColin Tan
    • CommentTimeJan 6th 2016

    Thanks Todd, that was helpful. I added to semicartesian monoidal category the motivating example ([0,],,+,0)([0,\infty], \ge, + , 0) of the extended real numbers.

    • CommentRowNumber6.
    • CommentAuthorColin Tan
    • CommentTimeJan 6th 2016
    In a semicartesian monoidal category, is it automatic that the tensor product is symmetric?
    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 6th 2016

    Surely not.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 6th 2016
    • (edited Jan 6th 2016)

    The usual examples do have a symmetric monoidal product though. It made me pause to wonder whether the nLab definition is the standard one.

    Edit: I’ll retract my previous sentence; I’ll bet it’s quite standard. In fact for any monoidal category MM, the slice M/IM/I over the unit is semicartesian, and that should be a universal example. I added some words to that effect (which should be checked for accuracy).

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJan 7th 2016

    I kind of doubt that the notion of semicartesian monoidal category is commonly enough used for anything to be “standard”. But the observation about slices is nice!

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 7th 2016
    • (edited Jan 7th 2016)

    Thanks! Something I found a little amusing yesterday was to consider the specific case Vect/kVect/k where kk is the ground field for VectVect. Both VectVect and AffAff (affine spaces over kk, including the empty one) embed fully in this category (the latter as a monoidal subcategory). For VectVect the embedding is V(V,0:Vk)V \mapsto (V, 0: V \to k).

    The embedding of AffAff takes a little more time to spell out. It’s A(1A,π)A \mapsto (1 \sqcup A, \pi) where \sqcup is the coproduct in affine spaces (akin to a simplicial join), 11 is the terminal affine space, and π\pi is the composite of 1!:1A111 \sqcup !: 1 \sqcup A \to 1 \sqcup 1 with a natural identification μ:11k\mu: 1 \sqcup 1 \cong k. Both 1!1 \sqcup ! and μ\mu which are morphisms of AffAff may be regarded as morphisms of 1AffVect1 \downarrow Aff \simeq Vect (pointed affine spaces are vector spaces) if we let the first inclusion i 0:111i_0: 1 \to 1 \sqcup 1 be the pointing of 111 \sqcup 1 and 0:1k0: 1 \to k the pointing of kk and define μ\mu by μi 0=0\mu \circ i_0 = 0, μi 1=1\mu \circ i_1 = 1 (the element 1k1 \in k). (So μ\mu is like two ends of a meter stick used to set up coordinates on the line kk.)

    “Most” objects (V,f:Vk)(V, f: V \to k) of Vect/kVect/k are “inhabited” in the sense that the projection to the terminal is regular epic; this means f:Vkf: V \to k is epic. For such objects, a morphism (V,f)(W,g)(V, f) \to (W, g) determines and is uniquely determined by the affine map f 1(1)g 1(1)f^{-1}(1) \to g^{-1}(1) between the fibers over 1k1 \in k, and thus we identify the full subcategory of inhabited objects of Vect/kVect/k with the category of inhabited affine spaces.

  1. By the way, Coecke and Lal call semicartesian monoidal categories causal categories.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJan 8th 2016

    @Todd: Cute. And the objects that aren’t inhabited are exactly those in the image of VectVect, right? So VectAffVect\cup Aff is all of Vect/kVect/k, and VectAffVect\cap Aff is 0k0\to k, the zero vector space and the empty affine space. There are no maps from an inhabited affine space to a vector space, and there is only the zero map from a vector space to an affine space, so the two are almost disjoint.

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 8th 2016
    • (edited Jan 8th 2016)

    Yes, they are almost disjoint, except that you do have maps ϕ:(V,0)(W,g)\phi: (V, 0) \to (W, g) from a “vector space” to an “affine space” whenever ϕg=0\phi \circ g = 0.

    I think what we have in fact is Vect/kVect/k realized as a collage of vector spaces with inhabited affine spaces (I’ll denote that category by Aff +Aff_+). It’s the collage induced by the bimodule B:Vect op×Aff +SetB: Vect^{op} \times Aff_+ \to Set where B(V,A)=Vect(V,T(A))B(V, A) = Vect(V, T(A)); here TT is the functor which takes an affine space AA to its vector space of translations T(A)T(A), which we can also express as the composite

    Aff +Vect/kkerVect/0VectAff_+ \to Vect/k \stackrel{\ker}{\to} Vect/0 \simeq Vect

    where ker\ker is of course the pullback along 0k0 \to k. I guess you’d just call that (some version of) the cograph of TT.

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 10th 2016

    Ah, I see. Cute!

    I wonder if one might regard this as another argument against admitting the empty affine space.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 10th 2016

    If you think of an affine space as a (locally trivial) principal V-bundle over a point, then it should be inhabited (maybe this point has been made already elsewhere).

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 10th 2016

    @Mike: I was wondering the same thing.

    @David: It may not have been said in exactly those terms, but similar points have been made elsewhere (e.g., an affine space is the same as a torsor of its space of translations). And that kind of point has some validity and force, surely, but not enough (for me, anyway) to override the opposing feeling that affine spaces over a field should form a variety, should form a complete/cocomplete category, etc., and that we teach our students that the solution space to Ax = b may be empty, etc. etc. This might be another instance of the nice object / nice category dichotomy. (Speaking only for myself, I’m not very comfortable at present with the nLab’s leaning toward the “nice object” side of the debate in affine space.)

    In the end whichever notion of affine space we choose probably comes down to context and expediency, but I’d be happier to see the category of inhabited affine spaces arise organically from satisfying categorical considerations. Along lines vaguely analogous to Tom Leinster’s Café post on how he came to love the nerve construction, which among other things lays out organic categorical reasons for choosing the simplex category over the augmented simplex category. There is a vague sense within the current thread (which I suppose should go elsewhere than “closed monoidal category”) that the structure of the slice construction Vect/kVect/k, which arises by applying the coreflection of monoidal categories into semicartesian monoidal categories to (Vect,)(Vect, \otimes), might provide a clue to this, maybe via this observation about Vect/kVect/k also being a collage construction.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2016

    Let’s continue at affine space.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2016

    I have added the statement that in a closed monoidal category the tensor/hom adjunction isos always internalize. (here)

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 30th 2018

    Added lctvs with the inductive tensor product as an example

    diff, v37, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeDec 20th 2018

    added pointer to Borceux.

    Does Borceux ever mention compact closure? Or else, what would be a good textbook reference for compact closed categories?

    diff, v38, current

    • CommentRowNumber21.
    • CommentAuthorSeth Chaiken
    • CommentTimeMar 30th 2023
    • (edited Mar 30th 2023)

    Should we distinguish the definition we give as a defining a right closed monoidal category?
    The Wikipedia article //en.wikipedia.org/wiki/Closed_monoidal_category (https) mentions the left version under the first definition.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2023
    • (edited Mar 30th 2023)

    What you suggest does not quite type-check, and I think the statement in the entry was correct. But I have now edited the pragraph a little to harmonize the variable names, which may make it easier to spot what the left- and the right adjoints are doing:


    …such that for each object Y𝒞Y \,\in\, \mathcal{C}

    1. the functor

      ()Y:CC (-) \otimes Y \;\colon\; C \longrightarrow C

      of forming the tensor product with YY,

    2. has a right adjoint functor

      [Y,]:CC [Y,-] \;\colon\; C \longrightarrow C

      forming the internal-hom out of YY

    in that for all triples of objects X,Y,ZX,\, Y,\, Z there is a natural hom-isomorphism of the following form:

    Hom 𝒞(XY,Z)Hom 𝒞(X,[Y,Z]) Hom_{\mathcal{C}}(X \otimes Y,\, Z) \;\simeq\; Hom_{\mathcal{C}}\big(X, [Y,Z] \big)

    diff, v47, current

    • CommentRowNumber23.
    • CommentAuthorSeth Chaiken
    • CommentTimeMar 30th 2023

    Thanks for the improvement! I’m glad to help. I fixed a easy typo in your edit \maspto to \mapsto.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2023

    Okay, good. Thanks for fixing the typo.

    (Incidentally, that entry closed monoidal category deserves the attention of an energetic editor: So far it does not do much beyond re-stating the definition of internal homs, three times in a row :-/)

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2023

    touched wording and formatting of the first few examples (here)

    and added mentioning of the previously missing examples of presheaves, vector spaces, and chain complexes

    diff, v49, current

  2. Fix reference to unbound variable.

    Tuplanolla

    diff, v50, current

    • CommentRowNumber27.
    • CommentAuthorBryceClarke
    • CommentTimeOct 26th 2023

    Added the example that Cat has precisely two closed monoidal structures, as shown in:

    diff, v54, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2023

    deleting the pointer to

    here and moving it to symmetric closed monoidal category

    diff, v55, current

    • CommentRowNumber29.
    • CommentAuthorncfavier
    • CommentTime7 days ago

    Mentioned the result from MacLane that allows to conclude that the natural isomorphism is natural in three variables from just an adjunction for each object.

    diff, v56, current