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I started to greatly expand the entry module
The new toc now looks like this:
Idea
Basic idea
More general perspectives
Enriched presheaves
Stabilized overcategories
Details
Ordinary concept
In enriched category theory
Examples
Related concepts
I am in the process of adding more stuff:
in the subsection As stabilized overcategories the beginning of the proof of what is apparently originally Quillen's identification of modules with abelian group objects in the overcategory
in the subsection Examples: modules over rings more details on the standard definition of ring modules
in the subsection Examples: modules over simplicial rings some statement taken from John Francis' blog comment
added an example-section on G-sets as the basic example of the generalized notion of module in the sense of enriched category theory
We should be careful -- these guys at many places say that the infty-stable categories agree with A-infinity and other approaches only over fields in characteristics zero. I am not competent to judge, but I would be careful not to go back and forth freely.
I think modules (some say algebras) over monads qualify as an important generalization in this entry. We also lack the treatment of modules over monoids in nonstrict monoidal categories, to include the coherences in the definition. I may do this, but if you wait for this it will take a while (tomorrow my boss is returning and we have a paper to finish these days).
Following Zoran's suggestion I added a paragraph on algebras over a monad
Then I typed more details of the proof that .
Finally I rearranged the existing examples section: with the different notions being discussed, I thought it would be beter to have the examples collected in the subsections that disscuss the definition of which they are an example of.
There are analogues of this theorem (square zero extensions = abelian group objects in comma category) for Lie algebras and some other categories. See also the lecture notes (listed in the references) by Markl et al. One of the people who reintroduced expression module over monad instead of algebra over monad is Deligne in his influential article Categories Tannakiennes; for Grothendieck school this is a must: everything has to be done in relative point of view. The qcoh modules over a relative affine scheme are exactly modules over a monad in the qcoh of the base scheme which has some additional properties (including its own right adjoint).
As far as G-sets are concerned, one takes just the cartesian product with the group G and the modules over that monad are G-sets. This way one does not need enriched point of view, a usual monad will do.
I have one question about usage of [I,C] in codomain fibration and I in other categorical situations, e.g. in treatment of nonabelian cohomology. In one case I is directed on-groupoid category generated by one arrow 0 to 1; while in another I is a groupoid generated by arrows 0 to 1 and 1 to 0. So how can one justify that an interval is played by directed and sometimes by undirected category with two objects ?
hereby I am moving an old discussion box from the entry module to here:
[begin forwarded discussion]
+– {: .query}
An earlier version of this entry led to the following discussion.
Eric: The wikipedia page distinguishes left $R$-modules as covariant functors and right $R$-modules as contravariant functors. Is that distinction important?
John: Yes, very — but I didn’t have the energy to get into that yet. For any ring $R$ there’s a ring $R^{op}$ in which $x y$ is redefined to be $y x$. I defined a left $R$-module above; a right $R$-module is the same as a left $R^{op}$-module. Eventually we’ll have to discuss all this stuff, which becomes vastly more important when we start talking about bimodules. If we want to show off, we’ll do it all not just for rings, which are monoids in Ab, but more generally for monoids in any symmetric monoidal category. For any monoid $M$ in a symmetric monoidal category we can define a new monoid $M^{op}$, and we can define left and right $M$-modules, and a right $M$-module is the same as a left $M^{op}$-module.
Sridhar: Given that left modules on rings are the covariant functors while right modules on rings are the contravariant functors, why does the above definition of a module on a monoid make the left modules the contravariant functors and the right modules the covariant functors? Is this actually the conflicting convention?
Toby: One problem is that this mixes with the conventions that one adopts for composition. What one person thinks is left multiplication, another will think is right multiplication. I would rather talk about left/right modules for monoids or rings, then talk about covariant/contravariant functors from categories or additive categories.
=–
[end forwarded discussion]
I am working on brushing up and expanding the entry module, which has been quite a mess. So far I have
spelled out the full definition at Modules in a monoidal category
and then made the discussion of Module over a ring more systematically a special case of this.
Polished a bit more here and there. But this entry still deserves further editorial attention.
There is also a notion of module over an algebra over an operad$^{[1]}$, but I’m not sure how best to fit that in with the article. (I’d also like to think about the shape and size of the article.)
[1] There must be dozens of children’s songs like the one I’m thinking of: “there’s a flea on a fly / on the head of a frog / on a bump on a log / in a hole on the bottom of the sea”.
A link to module over an algebra over an operad is (well hidden) in the section Modules over higher and generalized algebras .
Yes, I agree, the entry still needs reworking. It’s too long, too much spread-out.
We have also model structure on modules over an algebra over an operad :-)
Not quite as long yet as the rhyme you quote. But getting close ;-)
At module I have added an Examples-section Examples of modules over a ring (just the basic examples). Also rearranged slightly to make the overall section structure cleaner.
added to the Idea-section of module a section Motivation for and role of of modules: generalized vector bundles with a bit of discussion of how to think of modules geoemtrically, and how that helps to explain their relevance.
I added a bit more on modules over a ring $R$ as $Ab$-enriched functors, including the observation (quoted in MacLane) that, in this interpretation, the coend of the usual tensor product over integers is the tensor product over $R$.
added a section Abelian groups with G-action as modules over the group ring
added a section Modules - As stabilized overcategories - Modules over a group.
I emphasized where this article covers modules over an (enriched) category.
Is there discussion on the oo-categorial version of modules elsewhere in the Lab? Perhaps a concrete example is G CW-complex.
As a specific example, consider what a oo-module over the discrete group object ${\mathbb{Z}}/2$ internal to ooGrpd ought to be (a homotopical version of real space. An ordinary module over the discrete group ${\mathbb{Z}}/2$ is a set with an involution $f$ (such that $f\circ f$ is the identity). Due to being unable to distinguish ${\mathbb{Z}}/2$ from $\Omega {\mathbb{R}}P^\infty$ under homotopy equivalences of group objects, a oo-module over the discrete group object ${\mathbb{Z}}/2$ should be a oo-groupoid with a endomap $f$ such that $f\circ f$ is homotopic to the identity. Has a general notion of oo-action by a group(oid) object internal to a Grothendieck (oo,1)-topos been stated elsewhere on the Lab?
A side remark: does KR theory make sense when the involution is only up to homotopy?
Sure, all higher categorical versions are always linked to from theitr 1-categorical counterpart entries.
Try module spectrum, (infinity,n)-module, (infinity,1)-module bundle, for instance.
And for actions, see infinity-action. Naturally.
I extracted parts of the page into separate entries modules over a monoid and modules over an enriched category in order to be able to link to them from dg-module.
added pointer to:
(where $R$-modules are called $R$-linear spaces)
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