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I have added in the Idea-section at pointed object a brief sentence that pointed objects are the algebras of the “maybe monad”.
We had talked about this a while back in order to show that pointed objects in a presentable category are again presentable. So there seems to be something a bit deeper going on here than the simplistic concepts involved may superficially suggest. Maybe we should eventually have a dedicated section on this.
okay, I have added a brief remark at pointed object – Monadicity
Is there a way to see the smash product of pointed objects on purely general abstract grounds from the fact that they are algebras over the $X \coprod(-)$-monad? Is there some general statement about monoidal structure canonically induced on EM-categories that would give this?
I think the general idea is that the monad $1 + -$ on a distributive category is a monoidal monad (or commutative monad). I haven’t gone through it carefully, but the constraint
$(1 + X) \times (1 + Y) \to 1 + (X \times Y),$when you expand the left side to $1 + X + Y + X \times Y$, takes the summands $X$ and $Y$ to $1$. Seems to make sense. Then the EM category of a monoidal monad carries a canonically defined monoidal structure.
Thanks, Todd! That sounds like it’s going to be excellent. So you are saying it looks plausible that there is a canonical tensor product on the EM-category of $\ast \coprod (-)$ and that it is the smash product of pointed objects? That would be neat.
I just went to have a look at monoidal monad_and _commutative monad. I spent some time adding a bunch of cross references (numbering for definitions, pointers to them, etc.) because I found the material a bit hard to navigate.
One statement at commutative monad seems important, namely
In fact, it may be shown that commutative Lawvere theories on Set are precisely the same things as (finitary) symmetric monoidal monad structures on (Set,×), as shown by Anders Kock. For more on this, see monoidal monad.
I don’t think the entry monoidal monad actually gives much more on this. But I’ll look at Kock’s article. I suppose it will work not just in $Set$ but in any topos?
That seems doubtful. What’s an algebraic theory over a general topos?
Why “over”?, we need algebras of the theory in some topos.
I suppose I have now identified the source of the theorem alluded to in the entry, I guess it’s this article here (citation details now added at commutative algebraic theory):
There are finitary monads over Grothendieck toposes that are not induced by any Lawvere theory. For instance, $\mathbf{Cat}$ is finitary monadic over $\mathbf{sSet}$. The correct generalisation necessarily involves some kind of internal Lawvere theory.
It seems the statement has nothing much to do with Lawvere theories. Here is MO discussion of the result that we are after.
That’s a different result. It holds in any reasonable symmetric monoidal closed category; I can prove it when the base has equalisers, the category of algebras has reflexive coequalisers, and the forgetful functor from algebras to the base sends regular epimorphisms to epimorphisms.
Maybe we were talking ast each other, dunno. I just want to confirm, see above, that the canonical tensor product on pointed objects in any topos, regarded as algebras for the maybe operad, is the smash product. It seems clear enough now. If you give me your favorite citation, I’ll cite that.
The statement highlighted in #5 is correct; it seems the only thing Zhen Lin is objecting to is the final sentence of #5, where the suggestion ought to remember to replace “Lawvere theory over $Set$” by (say) “finitary monad over a (topos) $E$” or the like. But I don’t think this is a big deal.
I don’t have a citation, unfortunately. I believe the result is due to Kock, but the three papers [1970, 1971, 1972] don’t actually contain an outright statement regarding the monoidal structure, instead concerning themselves with the existence of a closed structure in the sense of Eilenberg and Kelly. But see also this preprint.
The general formula for the induced tensor product is as explained on MO: given $\mathbb{T}$-algebras $(A, \alpha)$ and $(B, \beta)$, one constructs the coequaliser
$F (T A \otimes T B) \rightrightarrows F (A \otimes B) \rightarrow (A, \alpha) \otimes_\mathbb{T} (B, \beta)$where the parallel arrows are the composites
$F (T A \otimes T B) \to F (A \otimes T B) \to F T (A \otimes B) \to F (A \otimes B)$ $F (T A \otimes T B) \to F (T A \otimes B) \to F T (A \otimes B) \to F (A \otimes B)$and in the case of the “maybe monad” this is just
$1 + (1 + A) \times (1 + B) \rightrightarrows 1 + (A \times B) \to A \wedge B$as expected. The strength of the monad is the evident natural morphism $A \times (1 + B) \to A \times 1 + A \times B \to 1 + A \times B$ induced by distributivity.
Thanks. And thanks for the pointer to Seal’s preprint.
I have added to the entry
I think it’s often easier to think about the internal-hom that’s right adjoint to the tensor product. For an enriched monad $T$ on a closed monoidal category and $T$-algebras $A$ and $B$, there’s an evident equalizer that defines a subobject of $[A,B]$ consisting of the $T$-algebra maps, and commutativity of $T$ says in dual form that this equalizer inherits the structure of a $T$-algebra from $[A,B]$ (whose algebra structure is induced pointwise from $B$). It thereby yields a structure of closed category on $T Alg$, and the tensor product is just the left adjoint to that — it can be constructed using a coequalizer, but often this characterization makes it easier to see what it is concretely. In the case of pointed objects, the resulting subobject of $[A,B]$ just “consists of pointed maps”, and you can check easily that the smash product is left adjoint to that closed structure on pointed objects.
Re Urs #1:
Any slice or coslice category of an accessible category is again accessible. There is a direct proof not involving (co)monadicity. More generally, the 2-category of accessible categories (and accessible functors) has comma objects, and these are preserved by the forgetful functor. (I think the accessibility of (co)algebras for a (co)monad actually depends on this result.)
Hi Zhen Lin,
your messages here keep coming a bit out of the blue, as far as I am concerned,I am not sure what you want me to do here (some months after the above exchange, as it were).
Since you refer to #1: in #1 I wrote:
Maybe we should eventually have a dedicated section on this.
You are maybe volunteering to start such a section!? That would be a useful contribution.
I have added bunch of basic facts, lemmas and examples to pointed object. In the course, I have slightly reformatted existing material, giving it numbered Definition/Proposition-environments.
and I gave pointed mapping space its own little entry, for ease of reference.
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