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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeNov 25th 2018
• (edited Nov 25th 2018)

created this in part to help answer query from Beppe here on the forum. It is very stubby and needs more work! (I copied and pasted from a preprint so it is not optimised for the Lab.)

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeNov 25th 2018

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeNov 25th 2018

Can you give a hint as to why he thought they were “of interest”?

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeNov 25th 2018
• (edited Nov 25th 2018)

First point, David, is that it is Grace Orzech. She was a student of Mike Barr and he had been looking at general results on algebraic cohomology theories (monadic cohomology etc.). The following is from a TAC paper on modified categories of interest. I am copying and pasting:

Categories of interest were introduced in order to study properties of different algebraic categories and different algebras simultaneously. The idea comes from P. J. Higgins [Higgins, 1956] and the definition is due to M. Barr and G. Orzech [Orzech, 1972]. The categories of groups, modules over a ring, vector spaces, associative algebras, associative commutative algebras, Lie algebras, Leibniz algebras, alternative algebras, Poisson algebras, left-right non-commutative Poisson algebras are categories of interest [Orzech, 1972, Casas, Datuashvili and Ladra, 2009, Casas, Datuashvili and Ladra, 2014]. Note that the category of noncommutative Leibniz-Poisson algebras defined and studied in [Casas and Datuashvili, 2006] is not a category of interest.

Whether or not you find that ’of interest’ is of course a question of personal preference! (The quote is from Theory and Applications of Categories, Vol. 30, No. 25, 2015, pp. 882–908 which may be ’of interest’ independently of this entry.) They link into the semi-abelian category set up as in Borceux and Bourn’s book and hence into forms of categorical universal algebra.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2018

Seems like a rather narcissistic name to me, along with being totally non-descriptive of its actual meaning. Even laying aside the people who may be “interested” in categories like noncommutative Leibniz-Poisson algebras (evidently a non-empty set, probably including at least Casas and Datuashvili), there are also plenty of people who are “interested” in totally different kinds of categories, like toposes for example. Was it originally intended as a permanent definition or as only a nonce? Is there any alternative in the literature that we could use on the nLab?

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeNov 25th 2018
• (edited Nov 25th 2018)

Good point. I do not know of another title. On reading the original paper by Grace Orzech, it seems that she had no better way of referring to it as it was the class of categories of interest to that study and the name just stuck. This is a bit like ’a convenient category of spaces’.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 25th 2018

Nah, “convenient category of spaces” is perfectly descriptive.

Let’s rename the entry, for it’s title sounds like someone is trying to make fun of something. When I opened this thread I was perfectly expecting that it was about you removing a spam entry of that title.

These categories will have an evident name. Maybe “categories of universal algebras” or the like.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2018

“Convenient category of spaces” could be more descriptive — convenient for what purpose? to whom? — but it’s a little better. But more importantly, a convenient category of spaces is something we choose in order to not have to think about it (for most purposes any convenient of spaces is as good as any other), whereas a “category of interest” is, as far as I can see, a general notion like “abelian category” of which one might study many examples that behave differently.

Actually I don’t even understand the definition of “category of interest”. The page says that its theory “contains” a set $\Omega$ such that blah. Does that mean that these are a specified subset of the set of all operations in the theory, or that they are all the (generating) operations of the theory?

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2018

Does anyone have any idea why the link to semi-abelian category is not getting parsed?

• CommentRowNumber10.
• CommentAuthorTim_Porter
• CommentTimeNov 25th 2018
• (edited Nov 25th 2018)

Urs you say:

I don’t even understand the definition of “category of interest”.

Looking at it again I note that I said the definition was copied and pasted from some notes of mine and they were unpolished to say the least! I will refine the presentation because your point is very valid…. and will copy the refined version to my notes for safety!

I would suggest category of group-based universal algebras or similar as the idea is one has a forgetful functor to the category of groups, but the extra operations are not completely general being generated by certain ones of low ’arity’ bound by some equations.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeNov 25th 2018

Urs you say:

I don’t even understand the definition of “category of interest”.

Mike said that. But it’s true, besides fixing the bad title, the entry needs clarification and structure.

• CommentRowNumber12.
• CommentAuthorTim_Porter
• CommentTimeNov 25th 2018

Urs: I have implied that it was a stub, so please lay-off and let me reword from that draft. It would be a help, rather, if you gave an opinion of my suggestion for a new title.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2018

I think “category of group-based universal algebras” isn’t too bad.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeNov 25th 2018

Feel free to rename the page too.

• CommentRowNumber15.
• CommentAuthorTodd_Trimble
• CommentTimeNov 25th 2018

Looking over the entry, is it the same as a category of algebras for a Lawvere theory $T$ over the Lawvere theory of groups (meaning $T$ comes equipped with a finite product-preserving functor $Theory_{groups} \to T$)?

If so, then one thing we could say is that the theory is Mal’cev.

• CommentRowNumber16.
• CommentAuthorTim_Porter
• CommentTimeNov 26th 2018
• (edited Nov 26th 2018)

Todd: There are quite a few results form the proto-modular / semi-abelian category theorists relating to this and, perhaps in time, I will be able to add in more as I find good references to the results. I think the answer to your query it Yes. (I found a mention that all semi-abelian categories are Mal’tsev, so these categories of group-based universal algebras are Mal’tsev.)

In the same vein, Beppe has added some stuff into Omega-group which points out that Higgins notion of group with multiple operators leads to a protomodular category.

• CommentRowNumber17.
• CommentAuthorTim_Porter
• CommentTimeNov 26th 2018

As suggested I have changed the name.

• CommentRowNumber18.
• CommentAuthorbeppe
• CommentTimeNov 26th 2018
Hi.

Indeed, categories of interest, is a bad name, I agree. Moreover it has been also used other (related) contexts. For instance, Murray Gerstenhaber, in

M. Gerstenhaber, On the deformation of rings and algebras: II, Annals of Mathematics, Second Series, vol 84, 1966,

uses the term for referring to subcategories of (non necessarily associative) rings closed under kernels, cokernels and fibred products. Other related notions arose in the quest for a general setting for studying interpretations a la Yoneda of cohomology theories of groups and various kinds of algebras, i.e. in terms of (crossed) n-fold extensions.

On the other hand, I would like to point out the fact the those categories of interest defined by Orzech are rally of interest. For instance, they are strongly protomodular, (and hence strongly semi-abelian). This means that in such categories you can deal with internal actions in more or less the same way you do in the categories of groups, see for instance

G. Metere, A note on strong protomodularity, actions and quotients, Journal of Pure and Applied Algebra, 221 (2017)

Moreover they are actin accessible in the sense of Bourn-Janelidze, meaning that actions are not so far from being representable, allowing a Schreier-MacLane-like obstruction theory for the classification of extensions.

Best,

B.
• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeNov 26th 2018

I am intending to put those points in later. I will try to give a feeling why these categories are ’of interest’ in this general area.

• CommentRowNumber20.
• CommentAuthorMike Shulman
• CommentTimeNov 26th 2018

Todd: wouldn’t that be a theory under the theory of groups?

• CommentRowNumber21.
• CommentAuthorTodd_Trimble
• CommentTimeNov 26th 2018

(-: I was divided on this myself. Sometimes we say an algebra over a commutative ring $k$ is an algebra map $k \to A$. And for posets too we refer to down-sets rather than over-sets. Anyway, of course I see your point.

• CommentRowNumber22.
• CommentAuthorMike Shulman
• CommentTimeNov 26th 2018

I would say that a map $k\to A$ is an algebra over $k$, but a ring under $k$. That is, if we view it as something defined in the category of rings, then the general principle of “under” applies, whereas if we view it as a presentation of a different kind of thing (an “algebra”) then we can use a different word.

I guess posets are just weird.

• CommentRowNumber23.
• CommentAuthorMike Shulman
• CommentTimeNov 26th 2018

Hmm, I’ve never really thought about this before, but even for posets (where the spatial metaphor is definitely reversed), I almost exclusively use different prepositions: when $x\le y$ I would say that $x$ is below $y$ or that $y$ is above $x$, never that $x$ is under $y$ or that $y$ is over $x$.

This is all rather a digression though…