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In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural transformation:

$$ \sigma F, F \sigma : F \rightrightarrows F^2 $$

are equal. As shown by Kelly all kind of transfinite constructions can be conveniently formulated as iterations of well pointed endofunctors.

I would like to give the "correct" generalization of this to $\infty$-categories (where $\infty$-category means $(\infty,1)$, and more precisely quasicategories).

A natural way to do it is as follow:

Let $\mathbb{N}$ denotes the monoidal category whose objects are non-negative integers, the monoidal structure is given by addition and there is a (unique) morphism form $m$ to $n$ if and only if $m \leqslant n$.

Fact: A well pointed endofunctor on a category $C$ is the same as a monoidal functor $\mathbb{N} \rightarrow End(C)$. Where $End(C)$ denotes the monoidal category of endofunctors of $C$, monoidal for composition.

This suggest the following:

Definition: A well pointed endofunctor on an $\infty$-category $C$ is a monoidal functor:

$$ N(\mathbb{N}) \rightarrow End(C).$$

Where everything is defined following the definition of Jacob Lurie's Higher algebra, in particular $End(C)$ is the monoidal category of endofunctors of $C$ as constructed in section 4.7.1 of Higher Algebra.

This all well and nice, But I have the impression that it is possible to give a purely finitary version of this notion.

Attempted equivalent definition : A well pointed endofunctor on an $\infty$-category $C$ is the data of:

  • An endofunctor $F:C \rightarrow C$.
  • A natural transformation $\sigma: Id_C \rightarrow F$.
  • An equivalence $\theta: \sigma F \sim \sigma F$ as natural transformation from $F$ to $F^2$.
  • A coherence $3$-cell $\Omega$ constructed expressing the following: The $1$-cell $\sigma \otimes \sigma : Id \rightarrow F^2$ is naturally equivalent (without using $\theta$) to both the composite $(F \sigma) \circ \sigma$ and $(\sigma F) \circ \sigma$, hence $\theta$ "composed" with $\sigma$ induces a self equivalence of $\sigma \otimes \sigma$. One should have a $3$-cell $\Omega$ expressing that this self equivalence is the identity.

Somehow surprisingly, I havn't been able to find any other "higher coherences" that seemed to be needed, so this suggest the following question:

Are the two definition above indeed equivalent (in an appropriate higher categorical sense) ?

So basically that would be a "coherence theorem for well pointed endofunctor".

I convinced myself some time ago that it was indeed the case using a complicated and very combinatorial argument in a model structure. But that argument could easily have been flawed. Moreover I have the impression that there could be simpler proof of this, so I wanted to raise the question on MO to see if someone could come up with either a more elegant argument or an obvious coherence condition in higher dimension that I have missed...

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    $\begingroup$ Here's some evidence in favor of this rectification hypothesis. A well-pointed endofunctor $(F,\sigma,\dots)$ corresponds to an idempotent monad iff $\sigma F = F\sigma$ is an equivalence. According to HTT 5.2.7.4, this rectification hypothesis does hold in the case where $\sigma F = F\sigma$ is an equivalence. I learned this from 4.17 here. $\endgroup$
    – Tim Campion
    Commented Dec 8, 2019 at 19:36
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    $\begingroup$ This is related to the fact that an idempotent in the homotopy category of an $\infty$-category, plus one additional coherence datum, can always be lifted to a homotopy coherent idempotent -- this is HTT 4.4.5.20 in the new versions -- it does not appear in the published version on HTT. Supposedly this appears as 7.3.5.14 in older versions of HA, but it does not appear in newer versions of HA. $\endgroup$
    – Tim Campion
    Commented Feb 25, 2020 at 22:38
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    $\begingroup$ @TimCampion : I'm not sure I see the link. But if there is one it rather point to a negative result to my question. In the case of idempotent you can extend finite level of the coherence condition to a fully coherent idempotent but this extention is non-unique and do induce an equivalence of categories between fully coherent idempotent and "partly coherent idemponent". $\endgroup$
    – Simon Henry
    Commented Mar 1, 2020 at 0:47

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If $D$ is a stable monoidal $\infty$-category then the theory of Smith ideals tells you that if you have a monoidal functor $F: \mathbb{Z}_{\leq 0} \to D$ then the cofiber of the map $F(-1\to 0)$ is a monoid and every monoid arises this way.

If you take opposites this should tell you that a well pointed endofunctor must be at least as complicated as a comonad.

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    $\begingroup$ I'm afraid I don't understand what you are saying exactly. Are you claiming that there is correspondence between monoidal functors $\mathbb{N} \to M$ and comonoid objects in $M$ when $M$ is stable and monoidal? $\endgroup$
    – Simon Henry
    Commented Jul 13, 2023 at 14:49
  • $\begingroup$ I think I have all my arrows going the right way now. Did this help? I think this says that if you have a monoidal functor $F: \mathbb{N} \to M$ the fiber of $F(0\to 1)$ is a comonoid. $\endgroup$
    – Justin Hilburn
    Commented Jul 13, 2023 at 15:20

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