2
votes
0answers
102 views

Preimages of accessible full subcategories

My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories ...
3
votes
0answers
127 views

Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
11
votes
3answers
407 views

Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
11
votes
3answers
554 views

The interplay between certain aspects of interpretability, model theory and category theory

I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would ...
7
votes
0answers
181 views

Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...
3
votes
1answer
127 views

Sketches for categories of models of complete theories

In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories ...
3
votes
1answer
153 views

Completeness theorem via syntactic categories

The nLab says in its internal logic article that the Completeness Theorem can be proven via a ``generic model'' of the theory. The model is generic in the sense that the only things true of it are ...
6
votes
0answers
214 views

Is there Ultracoproduct-like construction for topological spaces in general?

In http://arxiv.org/pdf/math/9704205.pdf they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
3
votes
0answers
153 views

Is there a useful Galois connection between Languages and Grammars?

I've just beginning to learn logic and proof theory - and the following rather vague and perhaps ill-formed question occurred to me. Given an alphabet it's straightforward to construct the Language, ...
3
votes
1answer
338 views

Interpretations as morphisms

Model-theoretic interpretations seem to give rise to categories in which morphisms are not functions. To name just two small examples: the category of finite graphs with interpretations between them ...
3
votes
2answers
321 views

The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein: start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...
1
vote
1answer
391 views

A Dedekind (pseudo) finite set

Quoting the wiki:- a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. -. ...
7
votes
1answer
566 views

2nd Incompleteness and Model Theory

In the presence of Godel's Completeness Theorem, the 2nd Incompleteness Theorem has the following strictly model theoretic interpretation: if there exists any model at all of (say) ZFC, there also ...
5
votes
3answers
1k views

Formalizing “no junk, no confusion”

Goguen has popularized the initial algebra view of semantics via his "no junk, no confusion" slogan. By "no junk", he means that models of a theory presentation should not have unnecessary elements, ...
0
votes
1answer
427 views

Is theory with domain of interpretation in second order objects a First Order Theory?

Thank everybody for answering my previous questions: first, and second. Here I would like to ask about some important thing which I do not understand clearly. Is it necessary for theory to have ...
11
votes
1answer
620 views

How are the two natural ways to define ''the category of models of a first-order theory T'' related?

Background/Motivation: Inspired by an interesting question by Joel, I've been wondering about the relationship between two very natural ways to define the category of ''all models of T'' where T is a ...
13
votes
4answers
2k views

Category theory and model theory as “natural” counterparts

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory ...
12
votes
3answers
801 views

Can we recognize when a category is equivalent to the category of models of a first order theory?

Many of the most canonical early examples of categories arise as the collection of models of a fixed first order theory, with the related model-theoretic concept of homomorphism. For example, the ...
2
votes
4answers
453 views

Category of groups = Category of models of group theory?

Is the category of groups with group-homomorphisms the same as the category of models of group theory with elementary maps? If not so: why?
1
vote
3answers
328 views

Can infinite first-order categories be specified other than as categories of models?

I am glad to see that a general question like Is there a relationship between model theory and category theory? receives quite a lot attention and no down-votes for being too general and unspecific. ...
9
votes
5answers
2k views

Is there a relationship between model theory and category theory?

According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relation while in Universal Algebra we ...
30
votes
7answers
2k views

Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept. ...