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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5
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1answer
122 views

Is there a term for a morphism which is surjective on generalized points?

Let $C$ be a category. Recall that a morphism $f : a \to b$ in $C$ is said to be a monomorphism if, for any morphisms $g_1, g_2 : c \to a$, it is true that $f g_1 = f g_2$ implies $g_1 = g_2$. ...
4
votes
1answer
342 views

Category of isomorphism classes?

Is there such a thing as a category of isomorphism classes of, say, modules? First step in definining morphisms in such a category would be to identify two morphisms $f:M\rightarrow N$ and ...
16
votes
2answers
504 views

Atiyah's definitions of Topological Quantum Field Theory

According to Atiyah, a TQFT is a functor from the category of cobordisms to the category of vector spaces. How does this definition relate with the physics of quantum mechanics? What does the ...
8
votes
1answer
491 views

Various definitions of group action

Sorry for the long post but this is a personal piece of maths, and I needed to be more precise as possible. There exists a well known equivalence between the category of $G$-sets and the category of ...
3
votes
3answers
558 views

Adjoint functors preserve (co)products

Let $F:\mathscr{A} \to \mathscr{C}$ and $G:\mathscr{C} \to \mathscr{A}$ be an adjoint pair of functors. I am trying to show $G$ preserves products and $F$ preserves coproducts. So to start we ...
1
vote
1answer
100 views

Instantiate spaces in commutative diagram by “appropriate” elements - name of this idea?

I wonder whether the following concept has a name. Suppose you are given a commutative diagram $\mathcal C$, that we think of a small category where each hom-class (i.e. hom-set) consists of at most ...
22
votes
4answers
627 views

A bestiary about adjunctions

What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying ...
1
vote
2answers
643 views

Why is the pullback not just a cartesian product (modulo a relation)?

Why is the pullback not just defined as in the case of the category of sets http://en.wikipedia.org/wiki/Pullback_%28category_theory%29 Perhaps there are some issues with categories that are not ...
8
votes
1answer
411 views

What does **Ens** stand for?

Earlier someone was asking about the category "Ens" described in Categories for the Working Mathematician. My question is more basic: What does Ens stand for? Most of the categories have names that ...
10
votes
1answer
544 views

The categories Set and Ens

I've recently started reading Categories for the Working Mathematician and I'm a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like ...
1
vote
0answers
191 views

Why is in the category of pointed sets not every epimorphism a cokernel?

A question in Tennison's Sheaf Theory is about the category of pointed sets and its characteristics. I have that its zero object is given by $(\{x\},x)$ the kernel of $f\colon (A,a)\to (B,b)$ is ...
2
votes
3answers
456 views

Definition of the quotient $G/H$

my question concerns the definition of the quotient $G/H$ in the sense of category theory. The following definition was obtained from the lecture notes linked to at the end of the question(p.2) ...
6
votes
1answer
320 views

What do coherent topoi have to do with completeness?

There is a theorem of Deligne that a "coherent" topos (e.g. one on a site where all objects are quasi-compact and quasi-separated) has enough points (i.e. isomorphisms can be detected via geometric ...
17
votes
7answers
627 views

What's the deal with empty models in first-order logic?

Asaf's answer here reminded me of something that should have been bothering me ever since I learned about it, but which I had more or less forgotten about. In first-order logic, there is a convention ...
10
votes
1answer
545 views

Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has ...
11
votes
5answers
543 views

Maps in Opposite Categories

Given some category ${\mathcal C}$, then the opposite category will consist of the same objects with the morphisms "turned around." Given $f:A\rightarrow B$, for $A,B$ objects of ${\mathcal C}$, then ...
2
votes
2answers
160 views

Uniqueness of Kernels in Abelian Categories

Refering to Serge Lang's "Algebra" pp. 133-134 on Abelian Categories, the following is unclear to me. Let $Q$ be an additive category and $F\stackrel{f}\rightarrow E$ a morphism. Let $A ...
2
votes
1answer
509 views

Isomorphism on commutative diagrams of abelian groups

Consider the following commutative diagram of homomorphisms of abelian groups $$\begin{array} 00&\stackrel{f_1}{\longrightarrow}&A& \stackrel{f_2}{\longrightarrow}&B& ...
3
votes
1answer
224 views

How to use the adjoint functor theorem construct the coproduct in Grp?

Let G and H are two groups,I know that the coproduct of them is the free product,but how to get it from the adjoint functor theorem? And I also want to see some applications of the adjoint functor ...
3
votes
3answers
117 views

Maps that assign points to maps

Consider a set $X$ and a set $Y$. Once can the define a map from $X$ to $Y$ that assigns to each point in $X$ a point in $Y$. On the other hand, if $F(X,Y)$ denotes the set of all functions from $X$ ...
2
votes
1answer
132 views

Natural transformations and small categories

If $\mathcal{A}$ and $\mathcal{B}$ are small categories (i.e. objects are sets, not proper classes) and $F,G:\mathcal{A} \to \mathcal{B}$ are both (contra/co)variant functors, then the the ...
7
votes
2answers
557 views

Does the following property of the direct limit of a direct system follow from the axioms for a direct limit?

Question: Does it follow from the axioms for a direct limit that if $\mu_i(x_i)=0$ then there exists $j \geq i$ such that $\mu_{ij}(x_i)=0$? Definitions and notation: (Atiyah MacDonald, chapter 2, ...
2
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1answer
280 views

Theory of supercategories

Category Theory has enormous utility as language for expressing mathematics, both continuous and discrete. It allows beautiful and succinct expression of that else be clumsy and clutterized. One ...
0
votes
1answer
391 views

How to understand the “create limit”?

I find it is hard to understand the "create limit." (You can find it in Mac Lane's Categories for the working mathematician, P112; there it defines: "A functor $V:A→x$ creates limits for a functor ...
6
votes
1answer
170 views

Is there a universal property of $\text{Spec}(-)$?

I've heard it been said that the construction of Spec$R$ is a canonical way of taking the ring $A$ and producing a locally ringed space with $A$ as the ring of global sections. This is certainly ...
9
votes
1answer
197 views

Isomorphisms in the localization of a category

Let $\mathcal{C}$ be a (small) category, and $S \subset \mathcal{C}$ a class of morphisms in $\mathcal{C}$. Suppose $f$ is a morphism in $\mathcal{C}$ that becomes an isomorphism in the localization ...
0
votes
2answers
107 views

How we can extract a vector space structure from a category with one object?

How can we associate a vector space structure to a category with one object ? Is there a canonical way of doing this ?
7
votes
3answers
729 views

when does a functor map products into products?

Motivation: wikipedia claims, that in algebraic topology, there holds: $\pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y)$ and $\pi_1(X\vee Y)\cong\pi_1(X)\ast\pi_1(Y)$. A similar statement holds for ...
0
votes
2answers
152 views

Adjoint question in the Category

If $\langle F,G,φ \rangle : X \to A$ is adjunction with $G$ full and every unit $η_x$ a monic, then every $η_x$ is also epi. Some similar questions like this, maybe I do not catch the key, who ...
1
vote
2answers
157 views

What does having a bar on a manifold mean?

If $M$ is a manifold then what is denoted as $\overline{M}$? I am guessing that it means a reversal of orientation. Related to the above I would like to understand the following construction of a ...
1
vote
0answers
174 views

On a Characterization of Exact Functors

A well known result states that, if $F:C \rightarrow D$ is a covariant functor between categories which admit finite projective limits, then $F$ is left exact if and only if it preserves finite ...
0
votes
1answer
209 views

direct limit versus projective

I have to be sincere here and say that I still don't get the real difference between a product and a sum. if I have the trivial order on a set of objects in a category. then the direct limit is a sum ...
85
votes
5answers
5k views

In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
17
votes
2answers
564 views

Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
27
votes
4answers
2k views

Can someone explain the Yoneda Lemma to an applied mathematician?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
0
votes
1answer
72 views

$\mbox{Hom}(A,\mathbb{Z}) \ne 0$ if and only if $A$ has an infinite cyclic direct summand

If $A$ is an abelian group, then $\mbox{Hom}(A,\mathbb{Z}) \ne 0$ if and only if $A$ has an infinite cyclic direct summand. The hint is to use If $F$ is a free abelian group and $g:B \to F$ is a ...
1
vote
2answers
184 views

How to understand the graph in the category?

I am reading the Mac Lane"s book: Categories for the Working Mathematician now,but I do not understand the "graph" in it.What is the different from the graph in the category and from the module ...
4
votes
1answer
271 views

Some isomorphism conditions

Here I want to ask some true or false problems regarding isomorphisms (and if false, is there some extra condition to make it true). I do not what is the correct title for these problem. And I also ...
3
votes
1answer
161 views

Natural Isomorphism $\mbox{Hom}(\oplus_\alpha A_\alpha,G) \simeq \prod_\alpha \mbox{Hom}(A_\alpha,G)$

In his algebraic topology book, Hatcher makes the comment that there is a natural isomorphism $\mbox{Hom}(\oplus_\alpha A_\alpha,G) \simeq \prod_\alpha \mbox{Hom}(A_\alpha,G)$ It is not immediately ...
1
vote
1answer
286 views

Yoneda Lemma - $\mbox{Hom}(\mathbb{Z},G) \simeq G$ [duplicate]

Possible Duplicate: How to show that for any abelian group $G$, $\\text{Hom}(\\mathbb{Z},G)$ is isomorphic to $G$. Simple question - is it true that $\mbox{Hom}(\mathbb{Z},G) \simeq G$ ...
14
votes
3answers
2k views

What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...
2
votes
1answer
197 views

presheaf as a colimit of representables

It is well-known that any presheaf (for simplicity say we're talking about presheaves of sets on a topological space $X$) is a colimit of representable presheaves in a canonical way. This has been ...
2
votes
1answer
66 views

Characterization of “Smallness” of the category of coverings over a topological space

Fixed a connected topological space $X$ it's an exercise to show that, if $X$ admits a universal covering $Y \rightarrow X$, then the category $C$ of finite covering spaces of $X$ is small. I'm ...
6
votes
1answer
267 views

natural numbers as direct limit

I would like to know if it is possible to write the natural numbers $\mathbb N$ as an inductive limit of finite monoids such that one recovers the natural multiplication of the natural numbers. Of ...
2
votes
2answers
176 views

Finitely generated free group is a cogroup object in the category of groups

I am trying to show that every finitely generated free group is a cogroup object in the category of groups. (Note I believe that this is also true for non-finite free groups, but that is probably much ...
6
votes
1answer
162 views

Is there a standard category-theoretic way to express a loop or quasigroup?

The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation. ...
4
votes
1answer
553 views

Group objects in category of groups

I am trying to show that a group object in the category of Groups is an abelian group (which, to a beginner, reads as a peculiar statement!) So here is what I know: Let $\mathscr{C}$ be a category ...
33
votes
1answer
798 views

Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $\left(m-n\right) \mid \left(f(m)-f(n)\right)$

A long time back, I wondered what functions other than integer polynomials on $\mathbb{N}$ (or $\mathbb{Z}$) satisfied the property: $$\forall m,n: \left(m-n\right) \mid \left(f(m)-f(n)\right)$$ ...
27
votes
0answers
608 views

Applying Freyd-Mitchell's embedding theorem on large categories

One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category. This is not immediately clear, since only small abelian categories can be ...
12
votes
0answers
210 views

What limits do commute with pushouts in Set?

I know of the following two colimit/limit commutation results in the category of sets: Products commute with sifted colimits and finite limits commute with filtered colimits. Does someone know ...