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1. • CommentRowNumber1.
   • CommentAuthorIS
   • CommentTimeJun 25th 2017
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   Author: IS
   Format: TextHello, I was just wondering about the fact that several pages presumes objects to be unital (for exemples the page on cogebras/coalegras). I'm not sure about the fact that it may be really convenient for categorical purpose, but I personally fell like it's more confusing in general (for example, I think that "bigebras" are always assumed to be unital and counital, while cogebras are not. And for Frobenius algebras, I have no idea what the convention is...). Moreover, there is sometimes not a single word about the non unital case. Following Bourbaki tradition (and definitions, see "cogebra" in Algebra, chapter 3), I think pages should first defines "X", without unitality assumption, then "unital X". Even for cogebra, where unity is not only supposed to be a unity of algebra but also to be a cogebra morphism, I think definition should be first without unity assumption (and then definition of "unital cogebra" should include the cogebra compatibility) (here I diverge from Bourbaki definitions, where bigebras are always unital and counital). I benefit from this post to ask for a little help: I think Frobenius algebras are always commutative by definition. Is cocommutativity automatic or an other assumption? (I have read it to be automatic, but can't figure out how to prove it...) PS: I'm new here, sorry if it's not the good category for discussion/question already ask/other. I'm also a bad english speaker, so equally sorry if it's full of mistakes :(.

   Hello,
   
   I was just wondering about the fact that several pages presumes objects to be unital (for exemples the page on cogebras/coalegras). I'm not sure about the fact that it may be really convenient for categorical purpose, but I personally fell like it's more confusing in general (for example, I think that "bigebras" are always assumed to be unital and counital, while cogebras are not. And for Frobenius algebras, I have no idea what the convention is...). Moreover, there is sometimes not a single word about the non unital case.
   Following Bourbaki tradition (and definitions, see "cogebra" in Algebra, chapter 3), I think pages should first defines "X", without unitality assumption, then "unital X". Even for cogebra, where unity is not only supposed to be a unity of algebra but also to be a cogebra morphism, I think definition should be first without unity assumption (and then definition of "unital cogebra" should include the cogebra compatibility) (here I diverge from Bourbaki definitions, where bigebras are always unital and counital).
   
   I benefit from this post to ask for a little help: I think Frobenius algebras are always commutative by definition. Is cocommutativity automatic or an other assumption? (I have read it to be automatic, but can't figure out how to prove it...)
   
   PS: I'm new here, sorry if it's not the good category for discussion/question already ask/other.
   I'm also a bad english speaker, so equally sorry if it's full of mistakes :(.
2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 25th 2017
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   Author: Todd_Trimble
   Format: MarkdownItexI think the general tendency for category theorists, and I suppose for most of the nLab regular contributors, is that the unital notions are generally nicer to work with. You can see this already in the notion of category where each object has an identity. Which is not to say that the non-unital structures are unimportant (we have for example [[semicategory]], which has its uses), but that "we" would prefer the unital notion as the default. So, for us [[rings]] have identities (otherwise we have "[[rng]]"), and same goes for [[algebra]] and so on. But I'd say please feel invited to insert a parenthetical "(unital)" if you feel it would be less confusing for other readers. Your English is intelligible, and I don't recall an earlier such discussion, so don't worry about any of that. There is a search utility up top which can sometimes help track down earlier relevant discussions.

   I think the general tendency for category theorists, and I suppose for most of the nLab regular contributors, is that the unital notions are generally nicer to work with. You can see this already in the notion of category where each object has an identity. Which is not to say that the non-unital structures are unimportant (we have for example semicategory, which has its uses), but that “we” would prefer the unital notion as the default.

   So, for us rings have identities (otherwise we have “rng”), and same goes for algebra and so on.

   But I’d say please feel invited to insert a parenthetical “(unital)” if you feel it would be less confusing for other readers.

   Your English is intelligible, and I don’t recall an earlier such discussion, so don’t worry about any of that. There is a search utility up top which can sometimes help track down earlier relevant discussions.

3. • CommentRowNumber3.
   • CommentAuthorRodMcGuire
   • CommentTimeJun 25th 2017
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   Author: RodMcGuire
   Format: MarkdownItexcategorically, __bounded__ [[lattice]]s are nicer than those without a top and/or bottom - the units of meet and join. In the nLab "bounded" is often not stated but assumed such as in the article on [[Lat]] the category of lattices. Actually, the "bounded" qualifier should really be "__bibounded__" meaning the lattice has both a __bound__ and a __cobound__. "Bound" is unambiguous for a [[semilattice]] and since a lattice is just a bisemilattice the 2 bounds come from the 2 semilattices. However I haven't heard anyone insist that "bound" just really means say initial element. The pages linked above also consider using "bounded" to be default and adopting special terminology for unbounded structures such as __pseudolattices__ and __semipseudolattices__. Usages such as __bottomless lattice__ (subtractive definition) or the equivalent __lattice with top__ (additive) depend upon whether one assumes a lattice has bounds.

   categorically, bounded lattices are nicer than those without a top and/or bottom - the units of meet and join. In the nLab “bounded” is often not stated but assumed such as in the article on Lat the category of lattices.

   Actually, the “bounded” qualifier should really be “bibounded” meaning the lattice has both a bound and a cobound. “Bound” is unambiguous for a semilattice and since a lattice is just a bisemilattice the 2 bounds come from the 2 semilattices. However I haven’t heard anyone insist that “bound” just really means say initial element.

   The pages linked above also consider using “bounded” to be default and adopting special terminology for unbounded structures such as pseudolattices and semipseudolattices.

   Usages such as bottomless lattice (subtractive definition) or the equivalent lattice with top (additive) depend upon whether one assumes a lattice has bounds.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 25th 2017
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   Author: Mike Shulman
   Format: MarkdownItexYes, the usual categorical perspective is that it's odd to have $n$-ary operations only for $n\gt 0$. I don't think we should change that, but if necessary we could clarify at the first usage on a page that things are unital. The prefix "semi-" is also more generally usable to denote the lack of identities: semicategory, semisimplicial set, semiring, semialgebra, etc.

   Yes, the usual categorical perspective is that it’s odd to have $n$-ary operations only for $n\gt 0$. I don’t think we should change that, but if necessary we could clarify at the first usage on a page that things are unital. The prefix “semi-” is also more generally usable to denote the lack of identities: semicategory, semisimplicial set, semiring, semialgebra, etc.