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)

1
vote
1answer
30 views

Special adjoint functor theorem (proof)

I'm currently going through the proof of the $\mathbf {Special ~ adjoint ~ functor ~ theorem}$ (SAFT) in Saunders Mac Lane's "Categories for the Working Mathematician" and I'm having trouble with the ...
1
vote
1answer
52 views

Motivation behind the definition of equivalence of categories.

The background: The naive notion of isomorphism of categories is that: a functor $F:\mathscr{C}\rightarrow\mathscr{D}$ is an isomorphism if there exists $F^{-1}:\mathscr{D}\rightarrow\mathscr{C}$ ...
2
votes
1answer
56 views

List of common and uncommon categories

I want to learn more about the category of "super commutative" graded $k$-algebra, for instance, its coproduct. However, I couldn't find anything related material. So, am I be able to get access to ...
1
vote
0answers
38 views

Left adjoint functor to forgetful functor from C*-algebras to *-algebras category [closed]

Does exist left adjoint functor of forgetful functor from category of C*-algebras to category *-algebras?
2
votes
0answers
33 views

Properties of the complexification functor for Lie algebras

The complexification of Lie algebras determines a functor from real Lie algebras to complex Lie algebras, whose right adjoint is the restriction of scalars functor. Thus, we know that complexification ...
2
votes
0answers
23 views

Does universal enveloped C*-algebra is continuous functor?

Let $K$ is category of *-algebras that have next property: for each $x \in B$ (where $B$ is *-algebra) $\sup_{\pi - bounded}||\pi(x)|| < \infty$ where $\pi : B \to B(H)$ - is some bounded ...
2
votes
0answers
32 views

Graphs and Krull-Schmidt Theorem

The book "Abstract Analytic Number Theory" of Knopfmacher states a similarity between Fundamental Theorem of Arithmetic and categorical Krull-Schmidt Theorem. Essentially, it states that, in some ...
2
votes
1answer
42 views

Name of a particular category

I'd like to work with a certain category which seems classic to me, but I don't know its usual name. Let's define $$Ob(\mathcal{C}) = \{(Y,Y_1,Y_2,f) : Y = Y_1 \cup Y_2, f : Y_1 \to Y_2\},$$ where ...
4
votes
2answers
175 views

Flat Modules are Filtered Colimits of Free Modules

A result by Wraith and Blass states that every flat module is a filtered colimit of free modules (see nLab, Thm 1). I am wondering if this is simply a corollary of Yoneda's density theorem which ...
1
vote
1answer
30 views

What is the internal hom functor in the context of an internally projective object?

I am trying to understand the definition of an internally projective object from nLab. It says that an object $E$ of a topos $\mathcal{T}$ is called internally projective if the internal hom functor ...
6
votes
2answers
115 views

Why a sheaf is an object that permits to get global information from local one?

Is there somebody who can explain/show me why a sheaf is something that can permit us to move from the local to the global? An explanation for the layman would be fine. Usually I tend to abhor them, ...
5
votes
1answer
104 views

Is there an functor without an adjoint?

So I'm doing some research into category theory, and I don't know whether this is a trivial question or not so I'll ask it anyway. Which functors don't have left adjoints? I know there must be some, ...
1
vote
1answer
32 views

finite presentations of algebraic structures

I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators. I'm ...
2
votes
2answers
51 views

How, intuitively, does commuting with filtered colimits capture “smallness”?

Definition. A compact object is an object representing a copresheaf which commutes with filtered colimits. In algebraic categories, the compact objects are the finitely presented ones, so commuting ...
13
votes
3answers
211 views

What is the opposite category of $\operatorname{Top}$?

My question is rather imprecise and open to modification. I am not entirely sure what I am looking for but the question seemed interesting enough to ask: The opposite category of rings is the ...
3
votes
0answers
38 views

С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
4
votes
2answers
72 views

Is the tensor product of non-commutative algebras a colimit?

For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of ...
5
votes
1answer
54 views

Examples of Waldhausen categories.

Waldhausen's wS construction of K-theory assigns K-groups to an arbitrary small Waldhausen category, my main goal in reading this construction was to apply it to the case of exact categories with weak ...
3
votes
1answer
62 views

Effective equivalence relations in a topos

I have a question about Johnstone's proof (in either Topos Theory or the Elephant; the accounts are essentially the same, so far as I can tell) that internal equivalence relations in a topos are ...
5
votes
2answers
52 views

S,R bimodules subcategory of the category of S+R modules?

If $R$ and $S$ are commutative rings, then does the category $R \oplus S$-modules encompase the category of $(S,R)$-bimodules? I was thinking we can accomplish this by defining the action to be: ...
2
votes
2answers
57 views

Opposite category trivial example

I have noticed that the basic notion of opposite category puzzles more than one person. I have also read many complete and motivated answers, as well as read definitions on books by Awodey, MacLane, ...
4
votes
1answer
50 views

When are canonical maps between limits monomorphisms?

If $\mathbf{D}_1 \hookrightarrow \mathbf{D}_2$ is an inclusion of diagrams in a category $\mathbf{C}$, and $\mathbf{C}$ has $\varprojlim \mathbf{D}_1$ and $\varprojlim \mathbf{D}_2$, then the ...
4
votes
1answer
47 views

Concrete description of (co)limits in elementary toposes via internal language?

In the category of sets, limits and colimits can be concrete described respectively as subobjects of products and quotients of coproducts. It seems like these descriptions make sense in any ...
6
votes
0answers
39 views

Completeness of Total Space of Fibration

The forgetful functor from category of ringed spaces to the category of topological spaces $F\colon RS\to Top$ is a bifibration. The fiber over each topological space $X$ is equivalent to the opposite ...
2
votes
2answers
63 views

Lemma 1.3.11 of Categories & Sheaves. Having trouble proving $F_0$ is unique up to unique isomorphism.

Lemma 1.3.11. Consider a functor $F: \mathcal{C} \to \mathcal{C'}$ and a full subcategory $\mathcal{C}_0'$ of $\mathcal{C}'$ such that for each $X \in \mathcal{C}$, there exists $Y \in ...
1
vote
1answer
44 views

To prove something is a functor isn't it enough to prove that it commutes with composition?

The second thing you usually have to prove is that $F(\text{id}_X) = \text{id}_{FX}$ for all $X \in C$, where $F: C \to C'$ is the supposed functor. I think it's enough to just prove that $F$ ...
6
votes
3answers
133 views

Why is it bad to pick basis for a vector space?

Reading `This Week's Finds', http://math.ucr.edu/home/baez/week247.html, I'm informed that one should avoid picking coordinate systems and I'm unsure why that is the case. Any help on the matter is ...
3
votes
2answers
65 views

The monoid of integers is not free

I am reading the introductory lessons on Category Theory on wikiversity, and they discuss free monoids here: https://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Monoids At the bottom ...
8
votes
7answers
497 views

Morphisms in the category of rings

We know that in the category of (unitary) rings, $\mathbb{Z}$ is the itinial object, i.e. it is the only ring such that for each ring $A,$ there exists a unique ring homomorphism $f:\mathbb{Z} \to A$. ...
3
votes
2answers
465 views

Why are contravariant functors called contravariant?

I'm just now learning a bit of category theory, and there often seems to be a certain notion, like limits for instance, and if you inverse certain arrows, you obtain a co-object related to that notion ...
3
votes
1answer
47 views

Requirement for having a left-adjoint functor

Let $G : \mathcal{C} \to \mathcal{D}$ be a functor between two categories. Suppose for each object $D \in \mathcal{D}$ there is a $C \in \mathcal{C}$ that is the "best approximation" of $D$ in the ...
2
votes
0answers
36 views

Direct Limits of Vector Spaces: Confusion about Definition of Mappings Given

First, I give the definitions I am using for the question. They are essentially those found on the Wikipedia page concerning Direct Limits. Let $\{V_i\}_{i\in I}$ be a family of vector spaces in ...
8
votes
0answers
60 views

Categorical formulations of basic results and ideas from functional analysis?

I'm taking a first (undergrad) course on functional analysis. Though the material is nice, the approach seems very ad hoc and in a sense, near-sighted (?). I was wondering whether the/a big picture ...
3
votes
0answers
30 views

On the definition of 2-rigs

I am reading the nLab entry on 2-rigs. In its list of definitions, it says that a 2-rig category can be defined as a $Ab$-enriched category which is enriched monoidal. Why is the enrichment in $Ab$? ...
0
votes
0answers
22 views

Proving Infinitely Ascending Chain of Subobjects

How would you prove that an infinitely ascending chain of subobjects of an object $X$ in $\mathcal{C}$ is stationary given that only finitely many preorders in the chain that are not isomorphisms in ...
1
vote
2answers
39 views

Having trouble proving natural transformation horizontal composition equality of two formulas using a diagram.

Let $C, C', C''$ be three categories. Let $C \xrightarrow{F_1, F_2} C'$ and $C' \xrightarrow{G_1, G_2} C''$, be four functors, and let $\theta : F_1 \Rightarrow F_2, \ \lambda : G_1 \Rightarrow G_2$, ...
4
votes
2answers
57 views

Etymology of transpose of morphisms in an adjunction

Let $F: \mathbf{C} \to \mathbf{D}$ be left adjoint to $G : \mathbf{D} \to \mathbf{C}$, witnessed by the family of bijections between hom-sets, natural in objects $X, Y$: ...
3
votes
1answer
79 views

classifying space of a category

In his K-book, chapter IV, Weibel states the following as a “a straight forward application of Van Kampen's Theorem”: Lemma 3.4 Suppose that $T$ is a maximal tree in a small connected category ...
3
votes
1answer
62 views

How to define the Yoneda embedding

I'm missing in my lecture notes the definition of "Yoneda embedding". It starts by saying that for a category $\boldsymbol{A}$ the Yoneda embedding is a functor $$ \boldsymbol{A} \rightarrow ...
5
votes
3answers
122 views

How do I read this diagram?

I had my first category theory class today and the professor used these kind of diagrams, and terms like "the diagram commutes". I come from another university, and I have no idea what these kind of ...
0
votes
1answer
38 views

Matching the definition of hom-functor with how these are used when defining adjuncts

I have a problem matching the definition of a hom-functor (from nlab) with how this concept it used in the definition of adjunction (from nlab): The hom-functor is defined on $C^{\text{op}}\times C$, ...
-3
votes
1answer
31 views

Less than equal to (<=) relation as morphism between natural numbers (objects)

How can I prove that "less than equal to" relation is valid morphism between natural numbers in category theory i.e it composes associatively.
3
votes
1answer
63 views

Ring structure with underlying abelian group $G$

If we start with an abelian group $G$, a ring structure with underlying group $G$ is in particular a map $\mu: G \otimes G \to G$, which by adjunction defines a map $\tilde{\mu}:G \to ...
0
votes
0answers
55 views

An exact sequence of inverse systems of $R$-modules

Let $$0\longrightarrow \big\{A_n,f_{mn}\big\}_{m \leq n} \overset{\Phi}\longrightarrow \big\{B_n,g_{mn}\big\}_{m \leq n} \overset{\Psi}\longrightarrow \big\{C_n,h_{mn}\big\}_{m \leq n} ...
1
vote
1answer
47 views

Defining adjoint functors: What does “natural bijection” mean?

Take the following definition of adjunction from the nlab This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for ...
3
votes
2answers
44 views

$(R^{\oplus A})^{\oplus B} \approx R^{\oplus (A\times B)}$?

Let $R$ be a ring and let $A$ and $B$ be sets. How can I see the following isomorphism of $R$-modules $$(R^{\oplus A})^{\oplus B} \approx R^{\oplus (A\times B)}$$ as a consequence of the universal ...
1
vote
1answer
38 views

Defining Presheaves on Categories

I'm learning about sheaves and sheaf cohomology with the eventually goal of using these tools to study Riemann surfaces. My references are Forster's Lectures on Riemann Surfaces and Iverson's ...
1
vote
1answer
35 views

Square is homotopy Cartesian if horizontal maps are weak equivalences

This is probably trivial but I'm not the best with category theory. Let $M$ be a right proper model category (that is pullbacks of weak equivalences along fibrations are weak equivalences). The ...
1
vote
1answer
43 views

Classifying Binary Associativity as a Unary Property

I'm working on a combinatorial style problem, specifically enumerating binary functions $\mu(x,y): M \times M \rightarrow M, x \in M, y \in M$ that are associative (here $M$ can be thought of as some ...
4
votes
2answers
215 views

Comparing morphisms of algebraic structures and topology

Let $X$ and $Y$ be groups, it seems natural to me that a proper function (homomorphism) $\phi$ from $X$ to $Y$ has this property that $\phi \subset X \times Y$ is a subgroup. Is there a way to define ...