Universal Algebra

From MGSA

Universal algebra

Take a non-trivial ultraproduct of the field $\Z/p\Z$ over all primes $p$ . Is it a field? What is its characteristic? Is it algebraically closed? (Bergman)
Make "generators and relations" into a functor. What is its left adjoint? (Bergman)
What is a Galois connection? Give some examples. (Rieffel)
- Must a Galois connection arise from a relation? (Bergman)
Let R be a relation on X which is reflexive and symmetric, X a set. Use R to define a Galois connection between and . Show that the following are equivalent:
1. $X\subseteq X^*$ and $X$ is closed.
2. $X$ is the intersection of sets $A i$ maximal with respect to having $A_i \subseteq A_i^*$ . (Bergman)
Are there free fields? How about free characteristic zero fields? (Rhodes)
What is the relation between freeness and adjoints? (Rhodes)
Define limit and colimit. Describe pullback in terms of limits. How is the pullback constructed in the category of rings? (Bergman)

Define "representable functor" and give examples. (Bergman)
What can you say about the collection of all objects $X$ which represent a given representable functor? (Vojta)
Give an example of a non-representable functor.
Make "generators and relations" into a functor. What is its left adjoint? (Bergman)
Tell us in some kind of understandable way about Quillan's Theorems A and B. (Rhodes)
What do the theorems say for posets? (Rhodes)
What do they say for the one object category corresponding to a group $G$ ? (Rhodes)
Describe the Yoneda Embedding and prove that it is free. (Rhodes)
Define monomorphism, epimorphism, injection, surjection (where appropriate). How are these connected?
Give an example of an epimorphism in a category with underlying sets which is not surjective.
What are the epimorphisms in the category of Hausdorff topological spaces?
Give an example of a partially ordered set which is not a lattice. What is the modular law for lattices? Show that it holds for the lattice of submodules of an abelian group. (Lam)
- Does every lattice have a universal homomorphism into a modular lattice (Bergman)
Define subfunctor. Is there a subfunctor of the identity functor on Group such that the subobject of $\Z/4\Z$ is $2\Z/4\Z$ ? What about such that the subobject of $\Z\times \Z$ is $0\times\Z$ ? (Bergman)

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