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I'll be rather general here, but I do not think the question is imprecise.

It is known that given a cotriple $\bot:\mathcal{C}\to\mathcal{C}$ we can construct a canonical simplicial object of $\mathcal{C}$. In general, are there qualities of a simplicial object that indicate that it is in fact the one associated to some cotriple? Or more concretely, given a simplicial object, are there properties that it can satisfy so that we can construct a comonad, or is all of the necessary information lost in the construction of such an object?

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Okay this edit is made months later, because I realized that my question is perhaps not exactly well posed. What I really want to know, and still haven't made time to actually investigate (because I don't think it'd be hard to play with, from a homological algebra point of view), is an answer to the question: Given a a functor $F:C\to S(C)$ when is it induced by a comonad (where by $S(C)$ I just mean simplicial objects in $C$)? We have one direction of a correspondence that takes comonads to simplicial functors on a category (perhaps with some restriction of the domain of the functor). Is there a way of going back?

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  • $\begingroup$ Can you say precisely how you construct a simplicial object of C from a comonad on C? What you write isn't literally true: e.g. there is a unique comonad on the empty category C, but we can't construct a simplicial object of C from it, since there are no simplicial objects in C. Do you mean, for instance, to start with a comonad on C and an object of C? $\endgroup$
    – Tom Leinster
    Commented Oct 12, 2011 at 2:41
  • $\begingroup$ It sounds like you have the bar construction in mind, isn't that always acyclic? do you have other canonical examples in mind? $\endgroup$
    – Sean Tilson
    Commented Oct 12, 2011 at 3:43
  • $\begingroup$ Right, Sean, presumably he is thinking of some version of the bar construction - but since there are several similar things called the "bar construction", we need to know which. $\endgroup$
    – Tom Leinster
    Commented Oct 12, 2011 at 3:54
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    $\begingroup$ At the root of one such bar construction is the simple observation that the algebraist's $\Delta$ (the category of finite ordinals with monoidal product given by ordinal concatenation) is the initial strict monoidal category with a monoid. Equivalently, $\Delta^{op}$ is the free strict monoidal category with a comonoid. If $C$ is a category, then a comonad on $C$ is the same as a comonoid in the endofunctor category $[C, C]$, seen as a strict monoidal category with endofunctor composition as monoidal product. Thus, given a comonad on $C$, we get a unique strict monoidal functor (cont.) $\endgroup$
    – Todd Trimble
    Commented Oct 12, 2011 at 4:23
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    $\begingroup$ $\Delta^{op} \to [C, C]$ which classifies that comonad. So the question is: what sort of simplicial objects in $[C, C]$ are monoidal functors? Abstracting away from this, we might as well ask: if $D$ is a strict monoidal category, what sort of functors $\Delta \to D$ correspond to those induced from monoids? And I don't know how to answer that without referring to the chosen monoidal structure on $D$. You get different answers depending on which monoidal structure you choose. I don't think the question is nonsense, but it requires the ability to sniff out the intended monoidal structure. $\endgroup$
    – Todd Trimble
    Commented Oct 12, 2011 at 4:29

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