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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 10th 2010
• (edited Oct 14th 2012)

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

• Modules over rings
• Related concepts

• Vector bundles and sheaves of modules
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 10th 2010

I am in the process of adding more stuff:

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 10th 2010
• (edited Jan 10th 2010)

added an example-section on G-sets as the basic example of the generalized notion of module in the sense of enriched category theory

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJan 10th 2010

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.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeJan 10th 2010

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).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 11th 2010
• (edited Jan 11th 2010)

Following Zoran's suggestion I added a paragraph on algebras over a monad

Then I typed more details of the proof that $Ab(CRing/R) \simeq R Mod$.

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.

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeJan 11th 2010
• (edited Jan 11th 2010)

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 ?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 29th 2012

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]

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 1st 2012

I am working on brushing up and expanding the entry module, which has been quite a mess. So far I have

Polished a bit more here and there. But this entry still deserves further editorial attention.

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeSep 1st 2012
• (edited Sep 1st 2012)

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”.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 1st 2012

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 1st 2012

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 ;-)

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 5th 2012
• (edited Sep 5th 2012)

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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeSep 11th 2012

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.

• CommentRowNumber15.
• CommentAuthorzskoda
• CommentTimeOct 2nd 2012
• (edited Oct 2nd 2012)

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 $R$.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 14th 2012
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeOct 24th 2012
• CommentRowNumber18.
• CommentAuthorTobyBartels
• CommentTimeJul 14th 2014

• CommentRowNumber19.
• CommentAuthorColin Tan
• CommentTimeJul 19th 2014

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?

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeJul 19th 2014

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.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJul 19th 2014

And for actions, see infinity-action. Naturally.

• CommentRowNumber22.
• CommentTimeJan 6th 2015

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.

• CommentRowNumber23.
• CommentAuthorJ-B Vienney
• CommentTime6 days ago

Added notion of multiplicatively cancellative module over a rig.