Timeline for Can infinite first-order categories be specified other than as categories of models?
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22 events
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| Feb 10, 2010 at 6:11 | history | made wiki | Post Made Community Wiki by Harry Gindi | ||
| Jan 18, 2010 at 16:13 | comment | added | Pete L. Clark | Sorry, but "How can an infinite graph be given?" is a conversation-ender for me: I cannot interpret that question in a meaningful but nontrivial way within the context of mainstream mathematics. If you are taking some non-standard approach, then you need to be ten times clearer and more explicit about what your assumptions are. Also I find the style of your discourse to be puzzling: "by accident" is not a paraphrase of "it happens", and I never said "given an infinite graph" (I said similar things, but that's not what quotation marks mean.) Maybe someone else can help you out... | |
| Jan 18, 2010 at 16:05 | comment | added | Hans-Peter Stricker | An explicit construction of the Rado graph relies on identifying its nodes with the the natural numbers and applying a rule to check if there is an edge between x and y. | |
| Jan 18, 2010 at 15:50 | comment | added | Hans-Peter Stricker | But you said "given an infinite graph". How can an infinite graph be given? | |
| Jan 18, 2010 at 15:49 | comment | added | Hans-Peter Stricker | My question is more or less the same as this one: How can I specify a definite infinite structure - a graph, a poset, a group, a category - without assuming its objects (nodes/vertices) to have some "inner structure" on which the arrows to be defined (edges/operations) rely on? | |
| Jan 18, 2010 at 15:46 | comment | added | Pete L. Clark | I already did. There are infinite graphs. | |
| Jan 18, 2010 at 15:29 | comment | added | Hans-Peter Stricker | By "by accident" I wanted to paraphrase your "it happens". | |
| Jan 18, 2010 at 15:27 | comment | added | Hans-Peter Stricker | I certainly can make up such an example, but only a finite one. Can you give me an infinite one? | |
| Jan 18, 2010 at 15:26 | comment | added | Pete L. Clark | ...E.g., given any graph, you can build a(t least one) category in which the objects are the vertices and the morphisms are the edges. (If multiple edges run between the same vertex, you have some choice as to how to define the composition of morphisms.) But I don't understand what mathematical question you're really asking: terms like "by accident", "rely on" and "specify" are opaque to me in this context. | |
| Jan 18, 2010 at 15:23 | comment | added | Pete L. Clark | @HS: I'm not sure who you've quoted or in what context, but the quoted statement is not mathematically meaningful. Yes, to define a category you don't need a preconceived notion of a structure and a homomorphism of that structure: you just need some objects, morphisms between the objects, and a composition law on compatible pairs of morphisms satisfying a few axioms. You can certainly make up an example of such a thing that does not correspond to any obvious pre-existing "structure"... | |
| Jan 18, 2010 at 14:13 | comment | added | Hans-Peter Stricker | @Pete: Let me quote another category theorists, who says, "the arrows don't, categorically speaking, know that they are specific homomorphisms, they are just some arrows you've put on a bunch of objects". My question is: How can I put the arrows on a bunch of objects without having an "intended model" in mind? How can I by accident recover a category to be such-and-such without having specified it as such-and-such beforehand? | |
| Jan 18, 2010 at 13:05 | comment | added | Pete L. Clark | @HS: your question does not seem well-defined to me. The usual way to "specify" the category of groups is to say that its objects are groups and its morphisms are homomorphisms of groups. I would say that this does not "rely" on any model-theoretic notions whatsoever: one does not need to know any model theory in order to understand or make this definition (and probably the majority of people who work with this definition do not know or care about model theory). It happens that you can recover this category as the category of models of a certain theory: so what? | |
| Jan 18, 2010 at 12:58 | comment | added | Harry Gindi | I think I'm somewhat out of my depth and you should ask someone else. | |
| Jan 18, 2010 at 12:22 | comment | added | Hans-Peter Stricker | I want to ask, whether - let's say - the category of groups can be specified in an independent way, not relying directly or indirectly on the class of models of group theory. (Do I have to define "specify"? I ask this honestly.) | |
| Jan 18, 2010 at 12:10 | comment | added | Harry Gindi | I don't know what you're asking though. If you restrict yourself to algebraic theories, then your question is a tautology, because it follows by definition that every algebraic category is equivalent to the category of models of that theory | |
| Jan 18, 2010 at 11:54 | comment | added | Hans-Peter Stricker | @Harry: Can you give me an advice how I could cure my question in a MO-conform way: Would it be OK to strike through the remark about higher-order theories in the first question and restrict it to "algebraic or relational categories" (corresponding to "algebraic or relational theories"), ruling out higher-order theories like topology (in the first place, as I originally intended to do)? A corresponding question for higher-order theories could follow later. | |
| Jan 18, 2010 at 11:39 | comment | added | Andrej Bauer | Ah yes. But also note the parenthetical remark about higher-order theories. That threw me off. | |
| Jan 18, 2010 at 10:47 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Jan 18, 2010 at 10:41 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Jan 18, 2010 at 10:39 | comment | added | Harry Gindi | He asked about first order models with a signature, which are exactly varieties of algebras, all of which can be described by a monadic adjunction between the base category and the category we're looking at. | |
| Jan 18, 2010 at 10:31 | comment | added | Andrej Bauer | I think he's looking at general first-order theories, but your argument that topological spaces are not an algebraic category only shows that topological spaces cannot be axiomatized as an algebraic theory (i.e., operations + equational axioms). They could still be aximatized in a richer language, except I don't understand what richer language Hans has in mind. | |
| Jan 18, 2010 at 10:27 | history | answered | Harry Gindi | CC BY-SA 2.5 |