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added a reference which uses this terminology:
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).
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 $X$-pointed objects are known as the objects of an $X$-coslice category.
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.
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.
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”. )
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 $C$ is a monoidal category, then $C^{op}$ also acquires a monoidal structure, and so the slice $C/I$ (“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 $Vect/k$ (with its induced monoidal structure).
Huh, the last two comments converge on similar points (no pun originally intended).
No, you were right the first time: you want to take the free pointing on the monoidal unit $\mathrm{1\ast$ of $Set$ to get the correct result.
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 $C^{c/}$ might remind one of is that the forgetful functor $C^{c/} \to C$ is monadic (for the evident monad $M \coloneqq c + (-) \colon C \to C$); cf. the common notation $C^M$ for the category of algebras.
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