User tryingotounderstand - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:46:28Z http://mathoverflow.net/feeds/user/16393 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more Why Lawvere theories have finite products? and more tryingotounderstand 2011-07-12T18:48:49Z 2011-07-13T07:33:00Z <p>According to Wikipedia, a Lawvere theory consists of a small category $L$ with (strictly associative) finite products and a strict identity-on-objects functor $I:\aleph_0^\text{op}\rightarrow L$ preserving finite products.</p> <p>Why a Lawvere theory have n-products for any n finite? For example, why isn't a Lawvere theory for monoids a category T with four elements: $1$, $T$, $T^2$ and $T^3$, and morphisms $e:1 \to T$ and $*:T^2 \to T$ making the appropiate diagrams commute and such that they are products of each other as expected? ($T^3$ is needed in order to state these diagrams)</p> <p>Also, why do they usually use the 'free' maps between the desired objects to model and not just the operators that the desired family of algebras has?</p> http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more/70182#70182 Comment by tryingotounderstand tryingotounderstand 2011-07-13T16:31:16Z 2011-07-13T16:31:16Z Andrej, I quote from nLab article on Lawvere theories: &quot;Remark. For T a Lawvere theory, we are to think of the hom-set T(n,1) as the set of n-ary operations defined by the theory. For instance for T the theory of abelian groups, we have T(2,1)={+,−} and T(0,1)={0}.&quot; Why is T(2,1) just {+,-},? Wouldn't it include other 'unnamed' operations as well? http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more/70187#70187 Comment by tryingotounderstand tryingotounderstand 2011-07-13T06:41:01Z 2011-07-13T06:41:01Z Thanks, after reading Andrej's post, now (3) makes sense. http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more/70182#70182 Comment by tryingotounderstand tryingotounderstand 2011-07-13T06:38:53Z 2011-07-13T06:38:53Z Thank you, very clear. I was misunderstanding it all, not thinking in syntax at all, just thinking about the models. http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more Comment by tryingotounderstand tryingotounderstand 2011-07-12T20:25:29Z 2011-07-12T20:25:29Z But isn't the monoid described just by the functor that tells what's the set ($F T$), the operation ($F *$) and the neuter ($F e$)? http://mathoverflow.net/questions/70162/why-lawvere-theories-have-finite-products-and-more Comment by tryingotounderstand tryingotounderstand 2011-07-12T19:46:54Z 2011-07-12T19:46:54Z I think this is the point where I get confused. Why would I need a product of four things? If model for $T$ would be a functor $F$ from $T$ to $Set$, then the product would be $F * : F T^2 \to F T$. For example if $F T = \mathbb{Z}_n$, then $F * : \mathbb{Z}_n^2 \to \mathbb{Z}_n$ would be the product (+) of my monoid (Z_n, +, 0) in Set, wouldn't it?