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The following question eats my brain: The standard definition of a "category" and a list of examples following this definition confuses me. My question is simple:

(Q1)If someone write "the category of finite groups" what are the objects of this category? Surely, $\mathbb{Z}_6$ is in this category. What about the other instances of $\mathbb{Z}_6?$ What prohibits to add extra copies of $\mathbb{Z}_6$ into objects? There is no "equality" of objects on "objects" other than the "standard equality class" of the object which models its theory. But the theory of an "object" is not included in the category. Let us formalize the theory of $\mathbb{Z}_6$ groups by adding extra conditions to the axioms of groups so that if a usual group satisfies these extra conditions then its isomorphic to $\mathbb{Z}_6.$ Now, is every model of these abstract conditions in our category? Or not? Do you include model theoretic semantic into category or not?

What does category theory give different than the model theory then?

(Q2) If someone chooses objects up to isomorphisms then why a "functor" (should be morphism) is called isomorphism? I saw somewhere that there is a mono which is not a monomorphism in the usual sense. So are there , for instance finite groups, which are isomorphic in the categoric sense but not isomorphic in the normal sense? Please note that finite group category is just an example, you are welcome to add interesting examples.

Thank you.

Edit: "functor" of Q2 should be "morphism".

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    I don't understand Q2.2011-07-05
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    You may want to have a look at [Lawvere's thesis](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html).2011-07-05
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    @Theo: thank you for the link. But could you also clarify how does this thesis related to the question? Is it for Q2? Could you also answer Q1 please? When do people say "category of groups" do they take iso. classes? Why they don't take 2 copies of the same group?2011-07-05
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    Two things: First: There is a *categorical* description of monomorphisms: $m$ is a monomorphism if $mf = m g$ implies that $f = g$. Now it turns out that even if $m$ is a map of underlying sets a monomorphism $m$ need not be injective. Dually there are epimorphisms ($fe = ge$ implies that $f = g$, which are a bit easier to come up with: e.g. the inclusion $\mathbb{Z} \to \mathbb{Q}$ is an *epimorphism* in the category of rings, or a map between topological spaces is an epimorphism if and only if it has dense range. Second: I gave you that link because Lawvere discusses among other things...2011-07-06
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    ...how different axioms of algebraic structures give rise to different categories (which mostly turn out to be equivalent). It may not be the best thing to read at this point, though.2011-07-06
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    Why is the title "a basic question (...)" when you ask two questions? In fact, the whole title is an unneeded extension of the tag. I'd put ideas from both questions.2013-08-30

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The natural notion of isomorphism of categories is not isomorphism (the existence of two functors which are inverses) but equivalence. A good intuition for equivalence is that it behaves like homotopy equivalence of spaces. In particular, just as many different spaces with different sets of points (in particular, sets of points with different cardinalities) can be homotopy equivalent, many categories with different sets of objects can be equivalent. Thus in the category of categories, "the set of objects" is not well-defined as it is not invariant under equivalence, just as in the homotopy category, "the set of points" is not well-defined as it is not invariant under homotopy equivalence.

When someone says "the category of groups," they are refraining from specifying a particular set of objects and morphisms because any reasonable choice gives the same category up to equivalence. For example, you can take

  • The category whose objects are sets $G$ (say in ZFC) equipped with maps such that etc. and whose morphisms are group homomorphisms, or
  • The category whose objects are, roughly speaking, isomorphism classes of groups and whose morphisms are group homomorphisms

and the corresponding categories are equivalent; the latter is just the skeleton of the former.

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    I will reread your post, but, it does not matter to take a copy of $D_4$ or two copies of $D_4$ in the objects. Is it correct?2011-07-05
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    @categoryboy: more or less. @Theo: okay, fair enough. I added a "roughly speaking" disclaimer.2011-07-05
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    Thank you very much for your answer. When people say "category of groups" do they take "iso classes of groups" or not?. What does "roughly" mean?2011-07-05
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    ok "skeleton" solves the question. I will reask Q2 later, namely the morphisms which don't have real world analogs later. Thank you.2011-07-05
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    @Qiaochu: Hmm, I'm not sure I believe your assertion about what happens when someone says "the category of groups". When *I* say "the category of groups" I mean the category whose objects are...groups! Namely, the first thing you say (or almost: I honestly don't think about sets in terms of ZFC set theory, although I might if pressed: in reality I am more Platonic / lazy [not in a good way!] than that). The second thing you describe leads, as you say, to a different and even non-isomorphic category. I don't think we should pretend that equivalent categories are actually *the same*, right?2011-07-06
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    Let augment my previous comment by saying that it is more than reasonable to only study properties of categories that are invariant under equivalence: indeed I think it is silly to do otherwise. (I guess the popular term is **evil**.) Similarly, I only care about properties of groups which are isomorphism-invariant, but it's not really coherent for me to pretend that the additive group of $\mathbb{F}_2$ is *the same* as the group of roots of unity in $\mathbb{Q}$, right?2011-07-06
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    In other words, go ahead and admit a (proper!) class of groups isomorphic to any given group in the category of groups: category theory handles this with perfect grace. Indeed, if I recall correctly some categorists view consideration of skeletal subcategories as verging on evil...2011-07-06
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    @Pete: well, all I am trying to say is that it shouldn't matter for any particular application which category you use up to equivalence. Even if you work in the skeleton, the same object of a category may enter an argument in different ways and that's perfectly fine, right?2011-07-06
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    @Qiaochu: right.2011-07-06
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    @Pete: As far as I can tell, Qiaochu is right that people don't mean a specific category (i.e., one determined up to isomorphism) when they say "the category of groups", at least if you restrict to people who are frequent users and "supporters" of category theory. This is just like when people comfortable with groups say "C_5$": they mean any cyclic group of order 5, not necessarily some specific model for it (such as congruence classes mod 5). But I remember that as a student I did want to know which specific group C_5 was, until I trusted the claim that it didn't matter.2011-07-11
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    @Omar: I'm not sure I agree with what you've said (although I have no mathematical problem with it). For instance, this is what the category *Groups* seems to mean in Mac Lane's book. Why would you want the category of groups not to be the category whose objects are precisely...groups? What is to be gained?2011-07-11
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    About $C_5$...that's a bit more subtle. I completely agree that **group theory** is the study of isomorphism-invariant properties of groups. But the basic object of study in group theory is still a group, not a group-up-to-isomorphism. For instance, what is the automorphism group of a group-up-to-isomorphism? Replacing objects by isomorphism classes of objects is actually a *decategorification*.2011-07-11
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    Even with $C_5$, people often work with one copy at a time: for instance one often wants to think of $C_n$ as $\langle x \ | \ x^n = 1 \rangle$, i.e., one wants to make reference to a distinguished generator. But groups-up-to-isomorphism do not have distinguished generators. Of course in the above presentation I don't care at all *what kind of object x is*, but I do want it to be some *thing*. So I don't care which specific group $C_5$ is either...this lack of caring is completely compatible with the viewpoint I am describing (and which I claim is rather standard).2011-07-11
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    @Pete: (1) Well, maybe not for the specific category of groups, but certainly categorically minded people have no problem in defining categories only up to equivalence. (2) I wasn't saying that people mean that $C_5$ is used for a group-up-to-isomorphism, but rather that it is used for a specific cyclic group of 5 elements, but without saying which one; i.e., it's not the isomorphism class of $\mathbb{Z}/5$, but rather some unspecified member of that isomorphism class. (I agree with everything you said by the way, I just want to clarify what I meant by the $C_5$ example.)2011-07-13
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    @Pete: are you sure you don't just secretly want to work in the category of group schemes instead? It seems like you want to say "but I need to distinguish $\mathbb{Z}/5\mathbb{Z}$ from $\mu_5$" or something like that, but of course there is no distinct entity called $\mu_5$ in the category of groups!2011-07-13
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    @Qiaochu: there are lots of distinct entities with different names corresponding to isomorphic groups: that's in the nature of my (non-skeletal) take on the category of groups. Thus for instance $\mathbb{Z}/5\mathbb{Z}$ is the additive group of a certain quotient ring of $\mathbb{Z}$, whereas $\mu_5$ is (or could be) the $5$-torsion subgroup of the multiplicative group of the complex numbers.2011-07-13
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    But your remark is still relevant: categories are (among other things) a formalism we use to work with mathematical objects. When I say $\mathbb{Z}/5\mathbb{Z}$, maybe I mean an object in the category of fields, maybe in the category of rings, maybe in the category of groups, maybe I mean a constant group scheme, and so forth. But anyway, for what (little) it's worth, $\mathbb{Z}/5\mathbb{Z}$ and $\mu_5$ are not completely indistinguishable groups: they're just isomorphic groups. Depending upon what you mean by $\mu_5$ they may even not be *canonically* isomorphic.2011-07-13
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Necessary context: In my opinion, it is not obligatory to base mathematics on set theory, although, of course, comparisons and discussions are interesting. Similarly, I do not think it is obligatory to compare categorical "foundations" with set-theoretic "foundations". At one end, Grothendieck found it necessary to postulate the existence of very many large cardinals (see Weibel's "Homological..."). At another end, in fact, by this point I definitely do not think that set theory reflects the practice of mathematics. Exaggerated category theory does not, either. I do not worry about "the category of sets", either.

The category of finite groups includes all finite groups... :) Yes, it includes many different copies of the dihedral group $D_4$, including different versions painted blue, red, or yellow. Or copies which are the same color, but are "distinct".

After some years/decades, I worry much less about the alleged set of sets that don't include themselves. I do not want a list of prohibitions that also prevents me from "forming" this set. I also do not want a prohibition against sharp knives when I am cutting up vegetables, even tho' I may cut myself. I do not want a prohibition against water, even tho' I may drown myself in my own bathtub by lying face down and breathing it in. Perversities.

So, yes, there are all those different copies of the same isomorphy-class of a group. Yes, if one thinks of ways to squeeze out a paradoxical-seeming something, probably one can. But why should one? :)

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    Is it just me or does your penultimate paragraph here have hints of paraconsistent logic? I'm not sure I can articulate this properly, but it seems that allowing such "[p]erversities" requires ignoring the contradictions they might create :)2014-03-08
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    Correction: "allowing such [p]erversities *in Mathematics*".2014-03-08
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    @Shaun, yes, in effect, I sincerely reserve the right to ignore some (seeming?) contradictions. My reference to "perversities" is that it is perverse to try to form the set of sets that don't contain themselves, for example, although it might have some sort of "academic interest".2014-03-08
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Q1: You indeed include all finite groups as objects. Any two copies of the same finite group are isomorphic, but not equal. Note that in the definition of a category, the class of objects is not required to be a set, but only the class of morphisms from one object to another is required to be a set.

You absolutely do not include any model theoretic notion whatsoever. Given a language and a theory, the category of models of that theory (with morphisms homomorphisms) is a category. That is the only relationship between these two notions.

Model theory and category theory are meant to express completely different things. It is true that both give a way to talk about mathematics, but for both this is not their ultimate role. Model theory is meant for one to express notions like provability, decidability, and to prove magnificently nontrivial theorems like the compactness theorem, and the Löwenheim-Skolem.

Category theory is mainly meant as a formal way of generalizing topological ideas. Thus, one gets extensions of the classical notion of homology, homotopy, and so forth. This is where it differ considerably from model theory: model theory is completely helpless when it comes to topology because the definition of a topology requires second order logic.

Q2: The definition of an isomorphism in a category is a morphism $r:A\rightarrow B$ that has a morphism $s:B\rightarrow A$ such that $r\circ s=id_B$ and $s\circ r=id_A$. In many cases, this coincides with your preconceived notion of what an isomorphism "should be" (indeed if what you want from your morphism to be an isomorphism for it just to be one to one and onto, then this is true). This is just a definition, and it only sometimes jibes with what you would want it to mean. As Qiauchu observed, in the category of categories isomorphism of categories is not what you really want to be considered an isomorphism. But life's tough that way. Same goes for mono and epic -- it often works with your intuition, but many times it doesn't.

You should note that not all categories are "concrete" (meaning that their objects are sets, and that their morphisms are functions between these sets): some may just be points with a bunch of arrows, and a rule for composition. So in these non-concrete categories, your intuition for what mono and epic mean (for example) has no meaning! So the categorical definitions have the advantage that they are very general.

Hope that helps in some way.

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    Thank you. Combining your first paragraph with "skeleton" of the category, we can use iso. classes or the collection of all groups as the "category of groups".(I am just verifying myself) Related to Q2, are there any iso s of a well-known category which is not "obtainable" by means of the usual techniques? Because I think it is possible to construct such a mono.2011-07-05
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    Regarding Q1 you are right, if you care about categories up equivalence rather than up to isomorphism of categories (I have never met anyone who cared about iso.'s of categories) then you can pick however many of any copy. Strictly speaking, when people say "all finite groups" you take all of them. About Q2: surely if you take a non-concrete category then there is no "usual technique". But for concrete examples (maybe someone can come up with an easier one) if you look at the category of topological spaces, with morphisms continuous maps where you identify any two homotopic ones, then an2011-07-06
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    isomorphism in that category will not imply a homeomorphism. But as you see from this example, the whole point is that "usual technique" is not a well defined term. What you're asking is: is a categorical isomorphism always what you expect? But you can expect many things. As I said in the answer, if you are in a concrete category (as groups) and you want to define an isomorphism as a morphism which is 1-1 and onto (as do want to define for groups) then the categorical isomorphism is indeed what you expect.2011-07-06
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    About topology requiring second-order logic, compare http://math.stackexchange.com/questions/46656/why-is-topology-nonfirstorderizable/46686#466862011-07-06
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    @Wesley Thank you Wesley for this wonderful info.2011-07-06
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    "Category theory is mainly meant as a formal way of generalizing topological ideas." This sentence sounds like something that Eilenberg or Mac Lane might have written circa 1955. In 2011 it sounds pretty old-fashioned: much of modern algebra uses category theory in a deep and essential way, for instance.2011-07-06
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    My description of model theory is also outdated (Shelah's work is far beyond my comprehension, for example). But those were the original applications of those theories. It was important to note that they are not competitors.2011-07-06