Timeline for How to define a generating subset for algebra in a category?
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2014 at 18:29 | comment | added | Qiaochu Yuan | @Adam: I don't think that convention is universal. | |
| Jun 16, 2014 at 18:25 | comment | added | Adam Gal | @QiaochuYuan Just terminologically I got the impression (from Lurie) that the word monoid is reserved for the cartesian monoidal structure and anything else is an algebra. | |
| Jun 16, 2014 at 12:38 | vote | accept | Christian Fischmann | ||
| Jun 16, 2014 at 5:59 | answer | added | Qiaochu Yuan | timeline score: 5 | |
| Jun 16, 2014 at 5:40 | comment | added | Qiaochu Yuan | @Christian: I don't understand your comment about ideals. This doesn't reduce to the usual notion of generating set for an algebra in $\text{Vect}$, say. | |
| Jun 16, 2014 at 5:39 | comment | added | Qiaochu Yuan | @Adam: I'm not sure what you mean. You can consider monoids with respect to any monoidal structure, not necessarily cartesian (in fact not necessarily symmetric). For example a monoid in $(\text{Vect}, \otimes)$ is an algebra in the usual sense. | |
| Jun 16, 2014 at 4:24 | comment | added | Adam Gal | I thought monoid just means associative algebra in the cartesian monoidal structure. At least this is Lurie's terminology. | |
| Jun 15, 2014 at 17:43 | history | edited | Christian Fischmann | CC BY-SA 3.0 |
added 170 characters in body
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| Jun 15, 2014 at 17:32 | comment | added | Christian Fischmann | @Marcel Yes this is what I mean. I have changed the question accordingly. | |
| Jun 15, 2014 at 16:29 | comment | added | Marcel Bischoff | I forgot: "$m$ and $e$ fulfilling the obvious identities" | |
| Jun 15, 2014 at 16:22 | comment | added | Marcel Bischoff | I guess the op means an algebra object also called monoid, which is an object $A$ and a morphism $m:A\otimes A \to A$, the multiplication, and a morphism $e:1\to A$ with $1$, the unit. E.g. take the category of $\mathbb Z_n$ graded vector spaces, then the $\bigoplus_{i=0}^{n-1} [i]$ has the structure of an algebra object and I would say it is "generated" by the subobject $[1]$. | |
| Jun 15, 2014 at 15:49 | history | edited | Yuichiro Fujiwara | CC BY-SA 3.0 |
made \em into italics (which is suggested by Sanath Devalapurkar) plus minor grammar corrections (by me because making it italic alone may be a bit too minor)
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| S Jun 15, 2014 at 15:46 | history | suggested | user62675 | CC BY-SA 3.0 |
made \em into italics
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| Jun 15, 2014 at 15:43 | review | Suggested edits | |||
| S Jun 15, 2014 at 15:46 | |||||
| Jun 15, 2014 at 2:47 | comment | added | Qiaochu Yuan | Anyway, a generating set should be a subobject $X$ such that the induced map $F(X) \to A$ is an epimorphism, or maybe a regular epimorphism, where $F(X)$ is the free algebra on $X$, whatever that happens to mean to you and provided that is well-defined. | |
| Jun 15, 2014 at 0:06 | comment | added | Qiaochu Yuan | Perhaps by "algebra" the OP means monoids, but in that case I don't know the relevance of the finite coproducts. | |
| Jun 14, 2014 at 20:53 | comment | added | Todd Trimble | Sorry, I don't know what you have in mind in the first sentence. This is an algebra in the sense of rings and algebras? I know how to define ring and algebra objects in categories with finite cartesian products, but what is the well-known definition you have in mind here? | |
| Jun 14, 2014 at 18:57 | history | edited | Christian Fischmann | CC BY-SA 3.0 |
added 282 characters in body
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| Jun 14, 2014 at 18:48 | history | asked | Christian Fischmann | CC BY-SA 3.0 |