Intereting Posts

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zero divisors of ${\bf Z}_n$
How to solve for $i$ and $n$ in compound interest formula?
Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary?
General and Simple Math Problem.
If any integer to the power of $x$ is integer, must $x$ be integer?
Orbit space of a free, proper G-action principal bundle
Arbitrarily discarding/cancelling Radians units when plugging angular speed into linear speed formula?
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
Closed form of the integral ${\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx$
covariant and contravariant components and change of basis
Tossing a coin with at least $k$ consecutive heads
Need help solving linear equations with elimination and substiution method
How to find the minimum value of $|5^{4m+3}-n^2 |$
Why is the disjoint union of a finite number of affine schemes an affine scheme?

I’ve done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I’m interested in how they might combine, particularly when applied to Algebra. [So far I’ve seen Lawvere Theories show up a lot.] What’s the best way to learn them both?

[EDIT 2: What books (if any) introduce them both in detail?]

Here’s my motivation: I’m fascinated by how different areas of Mathematics (and systems of Logic), particularly in Abstract Algebra, relate to one another and by how crucial certain axioms are in those relationships. I know this is quite vague but I’ve been chasing this stuff around for about a year; I know what it is I’m after when I see it, but I’m not knowledgeable enough (yet) to pin it down precisely. Category Theory & Model Theory (as well as Non-classical Logic) seem to hit the spot frequently.

- In an additive category, why is finite products the same as finite coproducts?
- Is there a category theory notion of the image of an axiom or predicate under a functor?
- Do random variables form a comma category?
- If a functor between categories of modules preserves injectivity and surjectivity, must it be exact?
- Hom-functor preserves pullbacks
- Relative chinese remainder theorem and the lattice of ideals

EDIT: To make things easier/more precise, think of a kind of “mathematical KerPlunk,” where the sticks are axioms and the marbles make up a (system of logic or) mathematical structure (with, say, red marbles for theorems, blue for definitions, etc.). If you remove (or change) certain sticks, what falls and why? Which marbles move? Do any change colour? Compare what you get with what you started with. How does the ‘new’ structure fit into the bigger picture? What are its ‘neighbouring structures’? That’s the kind of thing I’m interested in.

EDIT 3: Reverse Mathematics looks highly relevant but it’s new to me. Is there any way I could get there via CT & MT?

EDIT 4: Suggestions on how to improve this question are welcome. I think Universal Algebra is relevant but I’ve replaced its tag with the Topos Theory one to narrow things down.

- Fibered products in $\mathsf {Set}$
- Category Theory usage in Algebraic Topology
- Inductive vs projective limit of sequence of split surjections II
- how to show that a group is elementarily equivalent to the additive group of integers
- Why is there apparently no general notion of structure-homomorphism?
- Does taking the direct limit of chain complexes commute with taking homology?
- The free abelian group monad
- Categorical introduction to Algebra and Topology
- Inductive vs projective limit of sequence of split surjections
- Binomial rings closed under colimits?

I’ve actually taught myself a fair bit of category theory and am currently learning model theory. My own historical route is probably not what I’d recommend: Robert Goldblatt’s “Topoi” and Adamek et al. “Abstract and Concrete Categories”, followed by Lawvere’s “Sets for Mathematics” and Steve Awodey’s “Category Theory”. Probably doing that list entirely in reverse would be a gentler and saner approach.

I found the books on topos theory helped me a great deal, because I find set theory reasonably natural (ha) and it helped me relate categorical ideas to a category I had a fair sense for. I also made a project of doing all of Awodey’s exercises, which are accessible with a relatively minimal grasp of set theory and abstract algebra. Also, whenever I ran across an interesting theorem that I could understand the meaning of, I stopped and tried the proof myself.

For model theory I’m using a combination of Goldblatt’s book above, Chang & Keisler’s “Model Theory” 3rd ed. and David Marker’s “Model Theory: an Introduction.” Model theory doesn’t suit me as well as category theory, though, so I can’t really recommend how to approach the material. I can say that Marker’s a bit friendlier than Chang & Keisler. Multiple sources and doing lots of exercises doesn’t hurt.

Hope this gives you some leads.

- Prove that if $n^2$ is even then $n$ is even
- Inducing orientations on boundary manifolds
- $A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?
- Solve $2^x=x^2$
- Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$.
- convergence of $ \sum_{n=1}^{\infty} (-1)^n \frac{2^n \sin ^{2n}x }{n } $
- Simple solving Skanavi book exercise: $\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}}$
- Integration of x^(1/2) sinx
- Determine if a point is inside a subtriangle by its barycentric coordinates
- What is the predual of $L^1$
- Notation for modules.
- Characteristic subgroups $\phi(H) \subseteq H$
- Inverse images and $\sigma$-algebras
- Find the range of the given function $f$
- How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?