Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology newpage noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthoramarh
    • CommentTimeFeb 11th 2018
    Lax monoidal and (lax) closed functors between closed monoidal categories are essentially the same thing, and there is plenty of examples of strong monoidal functors that are only lax closed. However, I have been struggling to think of an example of a strong closed functor between closed monoidal categories that is not strong monoidal.

    Has anyone here encountered one of these "in nature"?

    I can see why the first situation would be much more common with cartesian closed categories, as "preserving finite products" happens under simple conditions, but I don't see why one should be more common than the other with general monoidal closed categories.
    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 11th 2018

    So in my book, strong monoidal implies preservation of the monoidal unit up to coherent isomorphism. In that case, wouldn’t the forgetful functor Vect AbVect_\mathbb{Q} \to Ab be an example?

    However, if we drop the condition of preservation of the monoidal unit and just ask that the structural constraint FAFBF(AB)F A \otimes F B \to F(A \otimes B) be an isomorphism, then I’m also struggling.

    • CommentRowNumber3.
    • CommentAuthoramarh
    • CommentTimeFeb 11th 2018

    Yes, to be clear: I am looking for functors between closed monoidal categories that

    • preserve the unit up to coherent isomorphism,
    • preserve internal homs up to coherent isomorphism,
    • do not preserve monoidal products.

    If we drop the first constraint, it is also not hard to find examples (for example, multiplication by an element gg in a group GG, seen as a monoidal category with only equalities as morphisms).

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 11th 2018

    Oh, I see: preservation of the unit is also implied by being strong closed. Got it.

    • CommentRowNumber5.
    • CommentAuthoramarh
    • CommentTimeFeb 12th 2018
    • (edited Feb 12th 2018)

    I think I’ve found an example, although it is a bit contrived. Maybe it reminds you of something more natural.

    Take CC to be 2\mathbb{Z}_2 as a strict symmetric monoidal discrete category, that is, CC has only two objects 00 and 11, only identity morphisms, and 0+0=1+1=00 + 0 = 1 + 1 = 0, 0+1=10 + 1 = 1.

    Take DD to be the strict symmetric monoidal poset with natural numbers as objects, sum as monoidal product, and a single morphism k+2nkk + 2n \to k for all k,n0k, n \geq 0. This is closed (I think) with [k,j]=jk[k,j] = j-k if jkj \geq k, 00 if k=j+2nk = j+2n, and 11 if k=j+2n+1k = j+2n+1. You can also see this as the “strictly commutative” PROP generated by a morphism 202 \to 0.

    Then the inclusion of 00 and 11 as objects of CC into DD is lax monoidal, with the morphism 202 \to 0 as the only non-identity structural morphism, but it is strictly closed.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 12th 2018

    I think a general class of examples should come from the inclusions of reflective exponential ideals. Day’s reflection theorem implies that if CC is a closed monoidal category and ECE\subseteq C is a reflective subcategory that is an exponential ideal in the monoidal sense (i.e. xCx\in C and yEy\in E imply [x,y]E[x,y]\in E), then EE is a closed monoidal category with the induced internal-hom and a reflected tensor product, so that its inclusion functor preserves internal-homs but is only lax monoidal (in contrast to its left adjoint, the reflection, which is strong monoidal). It won’t in general preserve the unit strongly, but I think there should be plenty of cases when it does.

    For instance, let C=PAC = P A be a presheaf category with a Day convolution monoidal structure induced by a monoidal structure on a small category AA, and let EE be the subcategory of Φ\Phi-continuous presheaves for some set Φ\Phi of colimits in AA that are preserved on both sides by the tensor product (e.g. if AA is itself closed). Then EE is a reflective exponential ideal, and contains the unit object since the latter is a representable presheaf (at the unit object of AA) hence preserves all colimits.

    • CommentRowNumber7.
    • CommentAuthoramarh
    • CommentTimeFeb 12th 2018

    Thanks Mike, that’s a great answer!

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 12th 2018

    I added it to the list of examples at closed functor.