Timeline for Are free functors usually injective up to isomorphism? [duplicate]
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 23, 2021 at 11:52 | comment | added | user1005113 | Thanks, Jeremy! | |
| Dec 23, 2021 at 10:24 | comment | added | Jeremy Rickard | @user1005113 I'm no expert, and won't embarrass myself by trying to give a formal definition, but my informal (and possibly wrong) understanding is that it refers to a way of making a model of $\tau$ from a model of $\sigma$ by defining the $\tau$-operations in terms of the $\sigma$-operations. For example, there is a homomorphism from the theory of Lie algebras to the theory of associative algebras since we can regard an associative algebra as a Lie algebra by defining $[x,y]:=xy-yx$. | |
| Dec 22, 2021 at 23:25 | comment | added | user1005113 | @JeremyRickard In your link, what is a homomorphism $\tau \to \sigma$ of finitary algebraic theories? (It is remarked that "if $τ→σ$ is a homomorphism and $σ$ satisfies IBN, then also $τ$ satisfies IBN".) | |
| Dec 22, 2021 at 19:34 | comment | added | Maxime Ramzi | I asked the same question here : math.stackexchange.com/questions/3325594/… - Eric Wofsey answered with the universal counterexample | |
| Dec 22, 2021 at 15:49 | comment | added | user1005113 | That's very helpful, thanks! | |
| Dec 22, 2021 at 15:43 | comment | added | Keith Kearnes | If the operations of the variety are finitary and the variety does not satisfy $x\approx y$, then it is the case that $F_V(X)\cong F_V(Y)$ implies $X\cong Y$ whenever $X$ is infinite. This is Theorem 1 of ``Note on the Isomorphism Problem for Free Algebraic Systems'' by Tsuyoshi Fujiwara, 1955. | |
| Dec 22, 2021 at 15:23 | history | closed |
მამუკა ჯიბლაძე Benjamin Steinberg CommunityBot |
Duplicate of IBN for algebraic theories | |
| Dec 22, 2021 at 15:21 | comment | added | Benjamin Steinberg | If a variety of universal algebras has a non trivial finite object then you do get injectvity. I don't know if this counts as usually | |
| Dec 22, 2021 at 15:11 | comment | added | Benjamin Steinberg | For Jonsson-Tarski algebras the free objects on any two finite sets are isomorphic | |
| Dec 22, 2021 at 14:54 | vote | accept | user1005113 | ||
| Dec 22, 2021 at 14:35 | answer | added | Jeremy Rickard | timeline score: 8 | |
| Dec 22, 2021 at 14:31 | comment | added | Jeremy Rickard | Related question: mathoverflow.net/questions/126747/ibn-for-algebraic-theories | |
| Dec 22, 2021 at 14:23 | answer | added | Tom Leinster | timeline score: 6 | |
| Dec 22, 2021 at 14:19 | comment | added | Mike Shulman | Correct me if I'm wrong, but isn't the terminal category a variety of algebras, for the theory with one nullary operation $p$ and one equation $\forall x (p=x)$? Its free functor is certainly not injective up to isomorphism. | |
| Dec 22, 2021 at 13:48 | comment | added | Yemon Choi | The abelianization functor from Grp to Ab is left adjoint to the forgetful functor, but is not injective on objects up to isomorphism. | |
| Dec 22, 2021 at 13:39 | history | asked | user1005113 | CC BY-SA 4.0 |