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 ...

learn more… | top users | synonyms (2)

6
votes
1answer
83 views

Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation?

Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows. $$(f+' g)(x) = f(x)+g(x)$$ Question. The ...
10
votes
1answer
426 views

Advice: Algebra and category theory for geometry?

I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some ...
7
votes
1answer
321 views

Isomorphisms between Normed Spaces

Between two normed (linear) spaces there are several notions of isomorphisms: Linear isomorphisms: linear bijective maps Topological isomorphisms: linear homeomorphisms (due to the linearity these ...
8
votes
1answer
330 views

Properties of Adjoint functors

Suppose that $(F,G)$ is an adjoint pair of covariant functors. I read in a book the folowing statement "if the right adjoint preserves epimorphisms, then the left adjoint preserves projectives." ...
10
votes
1answer
151 views

Zorn's lemma in categorical language

The axiom of choice, which is equivalent to Zorn's Lemmma, has a nice categorical "translation": in the category of sets, every epi is a retraction. So the axiom of choice says something about the ...
1
vote
2answers
103 views

kernel of a monic morphism

Problem Suppose $\mathscr{C}$ is an arbitrary category with zero object. $A$ and $B$ are two objects of $\mathscr{C}$. Let $f\in Mor_\mathscr{C}(A,B)$. It's given that $f$ is monic. I need to show ...
3
votes
1answer
65 views

How do morphisms in a comma category single out commuting squares?

I'm trying to teach myself the rudiments of Category Theory. I have a doubt about the definition of comma categories, more precisely about the morphisms. Suppose have two functors ...
5
votes
0answers
87 views

What is the benefit of the theory of categories? [duplicate]

I know the definition of a category and know several examples. But I have not studied the theory of categories. Much of mathematics I studied can be written as a category and a functor. But for me, ...
3
votes
1answer
79 views

In categorial logic, why do we need finite products to define the notion of “group,” but not “monoid”?

At n-lab (link), it says that ...the theory of groups lives naturally in the doctrine of categories with finite products, since a group object can be defined in any such category. Likewise, ...
3
votes
3answers
149 views

Understanding the Category of open subsets of a top. space X $Op_X$

I have a problem very similar to the one posted here by Brian. I guess he also stumbled across the introduction of the category of open subsets of a topological space X (denoted by $Op_X$) in Pierre ...
6
votes
0answers
171 views

A question about co-exponentials

An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$ or equivalently, as the terminal object of ...
2
votes
1answer
89 views

Proving the universal propetry of the coproduct for groups in $\mathrm{Ab}$

This question is from Alufi's algebra text. Show that if $G, H$ are abelian groups, then $G \times H$ satisfies the universal property for coproducts in $\mathrm{Ab}$. My attempt: Define ...
0
votes
1answer
70 views

Representability of transformations by morphisms

This is a very naive question. Suppose I'm given two functors $F,G\colon(Schemes/S)^{op}\to (Sets)$, over some base scheme $S$, represented by $S$-Schemes $X_F$ and $X_G$ respectively. There are ...
1
vote
0answers
30 views

Relations in points fibre are relations in base category

Let $\mathcal E $ be a category with finite limits and let consider the points fibre $\text {Pt}_Y $ over an object $ Y $ and the projection $\pi:\text {Pt}_Y \to \mathcal E$. The question is: ...
4
votes
3answers
139 views

Uniqueness of morphism (reasoning in categorial language).

This question is related to a previous question of mine. I figured out that maybe the right thing to ask for is for someone to solve one of the problems in Aluffi's book, that way I’ll know what the ...
2
votes
0answers
43 views

A full embedding in a finitely complete category

I know that each category can be full embedded in a complete quasicategory. My question is: a category can be full embedded in a finitely complete category? There is a universal such one (that's a ...
7
votes
2answers
165 views

How to get used to commutative diagrams? (the case of products).

I've returned to Aluffi's book (after getting the basics of groups from Herstein's) and I hit the same brick wall that made me put it aside. My general problem is this: I can't seem to get used to ...
5
votes
2answers
160 views

Conglomeration for conglomeration's sake

I know that category theorists have stretched the ontology of collections into conglomerates, 2-conglomerates, etc. My question is how far have they taken this? Are they interested at all in taking ...
5
votes
1answer
95 views

Does the existence of products in the category of sets imply the Axiom of Choice?

If for every family $(X_i)_{i\in I}$ of sets, there exists a categorical product in the category $\mathbf{Set}$ of sets, does this imply that the set-theoretic construction $\left\{(x_i)_{i\in ...
4
votes
1answer
38 views

Can this be proved purely on base of UMP?

Let $A,B$ be abelian groups and let $P$ serve as a product with projections $p_{A}:P\rightarrow A$ and $p_{B}:P\rightarrow B$. Let $C$ be an abelian group and let $f:C\rightarrow A$ and ...
2
votes
1answer
64 views

Is there a special name for ring homomorphisms $f : R \rightarrow S$ with $f^*(C(R)) \subseteq C(S)$?

Edit. For some reason, I called the functor $F$ described below a full functor as opposed to a faithful functor. The problem has now been corrected. For any ring $R$, let $C(R)$ denote the center of ...
6
votes
1answer
118 views

Lie algebra of Derivations as a functor?

To an associative algebra $A$ one can associate a Lie algebra $\operatorname{Der} A$ of all derivations $D:A\to A$. To any morphism of associative algebras $\alpha:A\to B$, how can one associate a ...
5
votes
1answer
226 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
1
vote
1answer
35 views

Does There Exist an Induced Model Strucutre via Ordinary Equivalence?

If $F:\mathcal{M} \to \mathcal{C}$ is an equivalence of categories, and $\mathcal{M}$ is a model category, does $\mathcal{C}$ inherit a model structure from $\mathcal{M}$ via $F$? If not, is there a ...
1
vote
1answer
57 views

Pointed objects in a category

Let $\mathscr{C}$ be a model category (following Mark Hovey's "Model Categories") and let $*$ be its terminal object. Consider the slice category $*/\mathscr{C}$: it's endowed with a canonical ...
9
votes
2answers
286 views

Introduction to sheaves using categorical approach

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful. For example, sheafification ...
0
votes
2answers
75 views

how to understand that products and coproducts are dual

I am reading some basic category notes, how can one relate the products to coproducts? If given a product, can one build its dual product? for example, the coordinate product $(x,y)$ : $ R \times R$, ...
3
votes
1answer
182 views

Right adjoint unique up to isomorphism

i want to prove the following without the Yoneda Lemma (because it is the exercise): Suppose $F\dashv G$ (with unit $\eta$ and counit $\epsilon$) and $F\dashv G'$ (with unit \eta' and conunit ...
6
votes
1answer
212 views

Can we capture all domains of discouse in the predicate logic within categorical logic?

In the construction of the bounded quantifiers via adjoints in the fibered category of subsets over a set (see e.g. here on Wikipedia), is there any restriction on the sets - specifically regarding ...
4
votes
2answers
89 views

Right derived functor and composition

Let $G:\mathscr{C}\to \mathscr{D}$ and $F:\mathscr{D}\to \mathscr{E}$ be functors, and suppose $F$ is right exact. It makes sense to me that in this case we have $$ R^i(F\circ G) \cong F \circ R^iG $$ ...
6
votes
1answer
193 views

Significance of unique isomorphism

In his answer to Unique up to unique isomorphism, Qiaochu Yuan explains quite well what is meant by something being "unique up to a unique isomorphism", but I'm a bit perplexed by the significance of ...
4
votes
2answers
100 views

What can we learn purely from the existence of a (non-constant) functor to the category of abelian groups?

I admit that the following is a very broad question. So if you feel that it is too vague please say so. It might also just be that I haven't read enough about category theory and my question is silly. ...
4
votes
1answer
88 views

Direct limit of topoloical spaces

Let $X$ be a topological space. Suppose $X_n$ are subspaces of X with $X_1 \subset X_2 \subset ... \subset X$. I'm going to prove $\varinjlim X_n =\cup_n X_n$. I have some trouble in proving that ...
15
votes
1answer
580 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
3
votes
1answer
82 views

Expressing the homomorphism condition with a commutative diagram.

I am asking myself how to express the homomorphic condition in commutative diagrams. For this let $(M, \cdot)$ and $(N, \cdot)$ be algebraic structures and let $\varphi : M \to N$ be a homomorphism, ...
3
votes
1answer
49 views

Closure and Co-Closure operations in the Pos category

In Simmons' "Introduction to Category Theory" exercise 1.3.6 he talks about two monotone maps either way between 2 Posets: ...
4
votes
3answers
184 views

Intuition or Motivation behind definition of Homomorphism - Fraleigh p. 29

p.29: A binary algebraic structure is a set $S$ together with a binary operation $*$ on $S$ and is denoted $<S, *>$ p.29: Let $<S,*>$ and $<S',*'>$ be binary algebraic ...
1
vote
1answer
91 views

Pullbacks and the power set functor

I have a very stupid question, so sorry in advance. I want to show that a power set functor $2^{-} : (Sets) \rightarrow (Sets)$ does not preserves pullbacks by a counterexample. Here is my logic: we ...
4
votes
1answer
79 views

The “closed” subspaces of topological algebraic structures

Every set-theoretic model of an algebraic theory gives rise to notion of (algebraically) "closed subset" in a canonical fashion; namely, the closed subsets are those that cannot be escaped via the ...
2
votes
1answer
35 views

Is a morphism of sheaves of abelian groups epi iff it is epi as a morphism of sheaves of sets?

Let $T=Sh^{Sets}(C)$ be the category of sheaves on a site $C$ and $S=Sh^{Ab}(C)$ be the category of abelian group objects in $T$, this is sheaves on $C$ with values in abelian groups. Is a morphism ...
-1
votes
1answer
61 views

How to define this function on arrows?

I want to solve the following exercise: Let $U: \mathcal{C}\rightarrow\mathcal{D}$ be a functor, $F:\operatorname{Obj}(\mathcal{D})\rightarrow \operatorname{Obj}(\mathcal{C})$ be a function. ...
1
vote
1answer
109 views

Proof that Beck-Chevalley holds for right adjoints iff it holds for left adjoints

I am looking at Bart Jacob's book "Categorical Logic and Type Theory". The proof of Lemma 1.9.7 is left as an exercise for the reader. It does not seem that easy to me, and i have had quite limited ...
8
votes
0answers
218 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
3
votes
1answer
47 views

Where can I learn more about these subcategories of functor categories?

Note: I've substantially edited the definition; $f$ is now allowed to be a functor. Given categories $\mathcal{C}$ and $\mathcal{D}$, we can form the functor category $[\mathcal{C},\mathcal{D}]$. Now ...
7
votes
1answer
101 views

Showing a functor that takes a group to its set of subgroups is not representable

Let $F:\text{Grp} \rightarrow \text{Set}$ be the functor that takes a group to its set of subgroups. Suppose $A$ is such a representing object, then $\text{Hom}(A,A) \simeq F(A) \Rightarrow ...
4
votes
1answer
109 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
2
votes
1answer
84 views

pull back and push out problem [duplicate]

could you please help with the proof? If a category $K$ has pullbacks. For composable morphisms $f: A\to B$ and $g: B \to C,$ if $g$ and $f$ are extremal epimorphisms, prove that $gf$ is an ...
3
votes
1answer
254 views

Is the forgetful functor from groups to monoids right adjoint?

I am working on the construction of a group free over a monoid. Maybe you know where I can find something about the (right) adjointness of forgetful functor $U:\mathbf{Grp}\rightarrow\mathbf{Mon}$. I ...
1
vote
0answers
47 views

Is there a connection between contravariant functors and the axiom of choice?

Given that both can be seen as talking about reversing arrows between two objects: Is there a connection between contravariant functors and the axiom of choice? I'm initially motivated geometrical ...
1
vote
0answers
50 views

Sufficient conditions for Kan extension's to exist and be flat/coflat and exist

I'm very new to category theory and a little confused about how to proceed with the following problem any help is welcome! Suppose $F:A \to B$ is a flat and coflat functor, that is; it preserves ...