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nLab > Latest Changes: pointed object in a monoidal category

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1 to 17 of 17

  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    Stub.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    added a paragraph on the category of generalized-pointed objects being the corresponding coslice category.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    added a reference which uses this terminology:

    • Paul-André Melliès, Nicolas Tabareau & Christine Tasson, p. 4 of: An Explicit Formula for the Free Exponential Modality of Linear Logic, in: Automata, Languages and Programming. ICALP 2009, Lecture Notes in Computer Science, 5556, Springer (2009) [doi:10.1007/978-3-642-02930-1_21]

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023

    Rather than a monoidal category, it suffices to consider pointed objects in a pointed category (i.e. having a distinguished object, not necessarily a zero object).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    Not sure what specifically this is referring to (zero object?), but it sounds like coming around to the point I have been raising a few times: If one goes full generality then XX-pointed objects are known as the objects of an XX-coslice category.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023
    • (edited Jun 5th 2023)

    Yes, it’s very general, which invites more discussion about what distinguishes monoidal unit-pointings from the rest of the crowd. One thing might be that there is an induced monoidal product on the category of monoidally pointed objects, just as in the classical case of pointed objects in cartesian monoidal categories (which has a cartesian product, as well as a smash product). In other words, the current entry invites more development and fleshing out. All in the fullness of time.

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeJun 5th 2023
    • (edited Jun 5th 2023)

    Not sure what specifically this is referring to (zero object?)

    The term pointed category is sometimes used to mean a category with a zero object.

    If one goes full generality then X-pointed objects are known as the objects of an X-coslice category.

    Though that’s true regardless of the generality. Arguably the fact that there is a canonical coslice category for a pointed category makes it worth distinguishing (at least conceptually) from coslice category, though, as we currently do for pointed object.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2023

    Incidentally, this is what Melliès (2009) notes:

    It is folklore that the category of pointed objects and pointed morphisms (defined in the expected way) is symmetric monoidal, and moreover affine in the sense that its monoidal unit 1 is terminal.

    (Rather than “folklore”, which usuall means “proof is thought to exist but hard to nail down in the literature”, probably he means “obvious”. )

    diff, v4, current

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    Returning to what I was saying in #6, one extra layer of conceptual flexibility that one has for monoidally-pointed objects (this terminology is growing on me!) is that if CC is a monoidal category, then C opC^{op} also acquires a monoidal structure, and so the slice C/IC/I (“monoidal copointing”) can be seen as a special case. And this is a source of fine examples. Witness for example the various interrelationships embodied between vector spaces and affine spaces, by considering Vect/kVect/k (with its induced monoidal structure).

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    Huh, the last two comments converge on similar points (no pun originally intended).

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2023

    Added something to the effect of my last comment to the Examples section.

    diff, v5, current

  12. added the example of bi-pointed sets, which are pointed objects in the monoidal category of pointed sets

    Joachim Joszef

    diff, v7, current

  13. brain fart, it’s the singleton not the boolean domain which is the tensor unit for the smash product.

    Joachim Joszef

    diff, v7, current

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 6th 2023

    No, you were right the first time: you want to take the free pointing on the monoidal unit *\mathrm{1\ast of SetSet to get the correct result.

  15. Fixed (again) the tensor unit for pointed sets

    Joachim Joszef

    diff, v7, current

  16. Use a more standard notation for the coslice category.

    Mark John Hopkins

    diff, v8, current

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 7th 2023
    • (edited Aug 7th 2023)

    I’m not opposed to the last edit, but you will find the other notation you replaced used all over the place. Urs uses it a lot, I believe. One thing that the superscript notation C c/C^{c/} might remind one of is that the forgetful functor C c/→CC^{c/} \to C is monadic (for the evident monad M≔c+(−):C→CM \coloneqq c + (-) \colon C \to C); cf. the common notation C MC^M for the category of algebras.

1 to 17 of 17

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