2
votes
0answers
107 views

Localization and Direct limit [migrated]

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
6
votes
2answers
358 views

Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone. In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion. Let's fix a representation $\bar{\rho}$ of a group ...
10
votes
0answers
179 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
3
votes
2answers
469 views

Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...
6
votes
1answer
370 views

Two questions about commutative theories

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
5
votes
5answers
2k views

motivation of filtered colimits

I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...
10
votes
5answers
5k views

Does category theory help understanding abstract algebra?

I'm studying category theory now as a "scientific initiation" (a program in Brazil where you study some subjects not commonly seen by a undergrad), but as I've never studied abstract algebra before, ...
4
votes
1answer
336 views

Mechanically instantiating abstract constructions

I am looking for work on the effective inverse of abstraction, aka specialization. There are two ways in which abstraction helps us: Get a better understanding of the structural rules at play in ...
16
votes
1answer
3k views

Infinite Tensor Products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
4
votes
2answers
774 views

when are epimorphisms of algebraic objects surjective?

let $C$ be the category of $\tau$-algebras for some type $\tau$. consider the statements: every monomorphism is regular. every epimorphism in C is surjective. it is easy to see that 1. implies 2. ...