Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
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2answers
212 views

$(-1)\otimes (-1) \cong I$

Is there a monoidal category $\mathcal C$ whose unit object is $I$ (i.e. $I\otimes A\cong A\cong A\otimes I$ for all $A\in \text{Ob}_\mathcal C$), with an object "$-1$" such that $$ ...
10
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2answers
514 views

Category Theory with and without Objects

Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an ...
14
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1answer
541 views

Categorification of $\pi$?

Is there a categorification of $\pi$? I have to admit that this is a very vague question. Somehow it is motivated by this recent MO question, which made me stare at some digits and somehow forgot my ...
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1answer
385 views

Pullback-stability for epimorphisms

In category theory, you see the idea of a class of epimorphisms being stable under pullback. For example, in a regular category, the class of regular epimorphisms is closed under pullback. Every ...
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1answer
398 views

A question in “Conceptual Mathematics”

In Conceptual Mathematics 1st edition, p. 325-236, there is a sketch of a proof, but I can't carry out the complete proof. "... This also follows from the appropriate universal mapping properties, ...
2
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1answer
188 views

Defining “subfunctors”

I have a functor $F\colon \mathbf{Rng}\to\mathbf{Grp}$, and a correspondence on objects which assigns to every group $F(R)$ a suitable subgroup $G_o(R)\subseteq F(R)$. Is there a way to turn $G$ into ...
5
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0answers
96 views

How to prove “The maps factoring through an injective object are precisely the null-homotopic maps”

Thanks for your attention, I'm an undergraduate. I'm reading the book of Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras, I cannot prove this ...
8
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1answer
351 views

Is $\mathbf{Grp}$ a concrete category?

Is $\mathbf{Grp}$ a concrete category? I thought it is, but then the group of symmetries of a square and the quaternion group are both of the order 8, and they are not isomorphic as groups. But sets ...
3
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4answers
316 views

Chromatic Filtration of Burnside Ring

I just attended a seminar on the chromatic filtration of the Burnside ring. I understood it relatively well, but at no point did anyone give an explicit definition of what a chromatic filtration ...
4
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0answers
99 views

Practical approaches to working with nonplanar commutative diagrams?

The 4-associahedron is the 4-dimensional version of Mac Lane's pentagon diagram. If you look at Trimble's notes on tetracategories, you can see the obvious difficulty in working with such a diagram ...
4
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2answers
255 views

Category of sets

Let C be a category of sets, which has objects all sets and arrows all functions, with usual identity functions and the usual composition of functions. For any set S, the assignment s-s for all s in S ...
2
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0answers
104 views

Field reductions. part three

Follow-up to Field reductions. part two For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting ...
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2answers
675 views

Why is Top a model category?

Recall that a model category is a complete and cocomplete category with classes of morphisms called cofibrations, fibrations, and weak equivalences. These are closed under composition and satisfy ...
6
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2answers
742 views

What is the difference between Categories and Relations?

For a common basis, I'll state basic definitions of a category and the relation type I'm thinking of. They're here for quick clarity, not precision, so feel free to revise for an answer. Category: A ...
5
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2answers
230 views

Where is the well-pointedness assumption of ETCS used in everyday math?

Where is the well-pointedeness assumption of the Elementary theory of the category of sets (Lawvere's category-theoretic axiomatization of set theory) used in everyday math? Specifically, if you have ...
3
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1answer
232 views

Is restriction of scalars a pullback?

I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback". I don't understand why this might be the case. In particular, consider a ...
4
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1answer
230 views

Yoga of localization in categories?

In the derived category $D(C)$ of an abelian category $C$, one formally inverts quasi-isomorphisms. In the context of model categories, one inverts weak equivalences. What does one gain by doing ...
15
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4answers
2k views

Category of all categories vs. Set of all sets

In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...
9
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2answers
423 views

Morphisms in the category of natural transformations?

I am learning the basics of category theory, so this question is probably obvious to anyone who knows the subject. The resources I've seen all take the following approach: 0) A category is a ...
10
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3answers
2k views

What are the prerequisites for learning category theory?

Is category theory worth learning for the sake of learning it? Can it be used in applied mathematics/probability? I am currently perusing Categories for the Working Mathematician by Mac Lane.
10
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1answer
347 views

Limits in the category of exact sequences

Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form $$ 0\to A\to B\to C\to 0 $$ and whose morphisms are triples of ...
2
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1answer
128 views

Is the empty subcategory thick, localizing, topologizing, etc

Let $A$ be an abelian category. There are various types of full subcategories. I often wonder if it is assumed that these are nonempty, since in most proofs this is used implicitely, but also the ...
5
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1answer
273 views

Why is the co-free module defined as the right adjoint to the forgetful functor to Ab rather than Set?

I'm currently reading Hilton & Stammbach's A First Course in Homological Algebra, and the following point has stumped me: In section 1.8, they construct co-free modules ("left moodule" over some ...
5
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2answers
740 views

Adjoint functors

I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor ...
4
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1answer
117 views

Do functions defined on global elements give rise to arrows in a well-pointed topos?

Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to global elements ...
2
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1answer
105 views

The name for a subobject(subgroup) which is annihilated by action

I know this question is easy, but for the life of me, I cannot remember what we call this thing. Googling for this has offered no help. Consider an object $A$ and a second object $B$(let them be ...
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1answer
240 views

Is the identity functor the terminal object of the category of endofunctors on C?

It seems to me not, since this would seem to imply that for all functors F and all objects A in C there exists a morphism F(A) -> A (making all functors co-pointed?). However, intuitively it seems ...
8
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1answer
278 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
5
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3answers
331 views

“Cat” modulo natural isomorphism?

I'm learning category theory by self-study. I have a couple of texts, and they both talk about how we ought to try not to think so much about the equality between objects in categories. Rather, the ...
13
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2answers
345 views

Finite generation in amalgamated free products

Let $G = A *_C B$ be an amalgamated free product of groups. My question is: suppose $C$ and $G$ are finitely generated, can we prove that so is $A$? I've been trying to prove it by contradiction. ...
31
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5answers
2k views

What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other ...
13
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696 views

what are the product and coproduct in the category of topological groups

I know the limits in the categories of groups, abelian groups and topological spaces and was wondering about the same thing.
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2answers
180 views

Can we turn the functor “category ring” into a 2-functor in a natural way?

Let $C$ be a small pre-additive category. Let $R(C)$ denote its category ring, that is, $$ R(C)=\bigoplus_{a,b\in \mathrm{Ob}(C)} C(a,b) $$ as Abelian group, where the direct sum runs over all object ...
6
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2answers
768 views

Learning to think categorically (localization of rings and modules)

I've been reading Ravi Vakil's notes on algebraic geometry notes and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal ...
0
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2answers
198 views

Existence of Natural Transformation between Functors

If F and G are functors between two arbitrary categories C and D, does a natural transformation η from F to G always exists? What is the condition for its existence? Thanks and regards!
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4answers
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What's more general than category theory?

First there was arithmetic with numerical calculations (i.e., one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then ...
1
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1answer
259 views

Complement of a submodule

Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q ...
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0answers
204 views

What categorical mathematical structure(s) best describe the space of “localized events” in “relational quantum mechanics”?

In a recent (and to me, very beautiful) paper, entitled "Relational EPR", Smerlak and Rovelli present a way of thinking about EPR which relies upon Rovelli's previously published work on relational ...
3
votes
3answers
341 views

Do endomaps of sets have interesting properties?

I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to itself?" Call ...
13
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4answers
572 views

Is there some universal sense of -ification (eg, groupification) in category theory

I have three questions. 1: Does the groupification of a semigroup always exist? I believe this should be yes because for every $x$ in the semigroup one could just define an element $x'$ that should ...
9
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3answers
324 views

Adjoint functors requiring a natural bijection

When showing that two functors $F:A\rightarrow B$ and $G:B\rightarrow A$ are adjoint, one defines a natural bijection $Mor(X,G(Y)) \rightarrow Mor(F(X),Y)$. What if one do not require the bijection to ...
4
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5answers
513 views

What's so special with small categories?

Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
4
votes
1answer
244 views

Colimit of $\frac{1}{n} \mathbb{Z}$

We should have $\displaystyle\mathbb{Q} = \lim_{\rightarrow} \frac 1n \mathbb{Z}$ but a few things are confusing me. Since the index category is a set, we should get the coproduct: $\bigsqcup \frac 1n ...
11
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7answers
843 views

Additive category that is not abelian

What is a simple example, without getting into the mess of triangulated categories, of an additive category that is not abelian?
6
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3answers
433 views

Lie algebras and infinitesimals

I have seen at many places the notions that Lie Algebras are infinitesimal objects and they look really close at a point. But I never understood this. They are abstract algebraic objects different ...
14
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2answers
2k views

Yoneda-Lemma as generalization of Cayley`s theorem?

I came across the statement, that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations. How exactly does generalizes ...
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1answer
203 views

Thin categories and graphs (isomorphism of categories)

I try to learn the theory of category from The Joy of Cats. I got stacked with the first exercise (3A a). If we have a simple graph with one vertex and 2 nodes all we know is that in category there ...
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5answers
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Constructing a counterexample in category theory

Exercise 10 in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). ...
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3answers
261 views

Isomorphism and $\mathrm{id}$

In a category I have two objects $a$ and $b$ and a morphism $m$ from $a$ to $b$ and one $n$ from $b$ to $a$. Is this always an isomorphism? Why is it emphasized that this has to be true, too: $m \circ ...