1
vote
0answers
57 views

Generators and relations as a functor [closed]

Make “generators and relations” into a functor. What is its left adjoint? [Bergman] How could one do this?
3
votes
0answers
29 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
2
votes
1answer
25 views

The functor of monoids

I'm studying this book on introductory level category theory and I couldn't solve this exercise: In the first part I've been thinking about the monoid homomorphisms $F: S\to T$ and regarding of ...
1
vote
1answer
62 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
2
votes
3answers
211 views

Definition of groups in a more abstract way

I'm trying to understand the definition of group objects in categories, this is an extract of Paolo Aluffi's book: QUESTIONS Can I say that $e(1)$ is the identity in our group $G$ we have just ...
9
votes
3answers
117 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...
2
votes
0answers
44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
1
vote
0answers
40 views

coalgebra/algebra of the identity endfunctor

Let $\mathbb{C}$ be a small/locally small category and let $T:\mathbb{C} \to \mathbb{C}$ be an endofunctor. One can then have $T$-algebras and $T$-coalgebras in the usual way: for $X,Y \in ...
2
votes
0answers
36 views

Push-out of product of diagrams

Im working on the category of groups. Let $D$ be the push-out of the diagram $B\leftarrow A \rightarrow C$. Let $D'$ be the push-out of the diagram $B' \leftarrow A' \rightarrow C'$. It is possible to ...
1
vote
2answers
32 views

Precomposition with a faithful functor

If $F: C \rightarrow D$ is a faithful functor, then is the precomposition with $F$ functor $F^{\star}:[D:\mathbf{Set}] \rightarrow [C:\mathbf{Set}]$ faithful?
5
votes
2answers
99 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
1
vote
1answer
38 views

Explanation of a natural transformation in group theoretic terms

In category theory, a functor looks like homomorphism between categories. Keeping that analogy in mind, can a natural transformation be described by (or restricted to) group theoretic terms? For ...
2
votes
2answers
54 views

Zero direct limit of nonzero objects

Can anyone present to me kindly a directed set of nonzero objects with the zero direct limit? I first tried $$F(U)=\{f:U \to R \mid f\text{ is continuous}\}$$ in p.507 of "Advanced Modern ...
1
vote
1answer
53 views

Do the circle groups have any interesting stand-alone descriptions?

By the circle groups, I mean firstly the circle group $\mathbb{T} \subseteq \mathbb{C}$ of all complex numbers having modulus $1$, and secondly the commutative group $\mathbb{S} = \mathbb{T} \cap ...
6
votes
3answers
171 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
0
votes
0answers
26 views

Resources for Polyadic and/or Cylindric Algebra

I'm looking to learn a little bit about polyadic and cylindric algebras, as part of an investigation into algebraic approaches to logic. The only "text" that I can find for polyadic algebra is ...
1
vote
1answer
46 views

The injective objects in the category of algebras

What is the definition of the injective objects in the category of algebras?
5
votes
1answer
90 views

Is there an algebraic invariant for complex curves that's mapped to injectively?

Consider the functor $\pi_1: \text{Closed Surfaces} \rightarrow \textbf{Grp}$. This is homotopy invariant; every closed topological surface has a unique fundamental group. In the reverse direction, by ...
0
votes
0answers
34 views

Is the functor category of algebras into modules locally small?

Let $R$ be a ring. Is $[{}_R Alg, {}_R Mod]$ locally small?
3
votes
1answer
37 views

Inducing a comodule structure on Hom

If C is an R-coalgebra and M is an R-module... then is it possible to endow $Hom_{_RMod}(C,M)$ or $Hom_{_RMod}(M,C)$ with the strucutre of a C-comodule?
7
votes
1answer
86 views

Should this be viewed as a serious issue with the meadow-theoretic approach?

Meadow theory (see here) allows us to apply the results and concepts of universal algebra to the study of fields. Obviously, this is very, very nice. However, I have the following issue with the ...
4
votes
2answers
75 views

Hom of algebras

For any $R$-algebras $A$ and $B$, doea their set of R-algebra morphisms $\mathrm{Hom}_{R_{\mathrm{Alg}}}(A,B)$ necessarily again have the strucutre of an $R$-algebra?
5
votes
1answer
126 views

What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
1
vote
1answer
29 views

Acyclic resolution but not projective

Suppose $\mathfrak{C}$ is an abelian category which does not have enough projectives and we're interested in computing the right derived functors of some covariant functor $F$. If however, every ...
0
votes
0answers
31 views

Comodules as a functor category

Let C be a comonoid in some preadditive monoidal category $\mathfrak{C}$, then how can we express the category of C-comodules, in terms of some sort of functor category? I mean is there a similar ...
2
votes
1answer
38 views

Cocompletion and functor categories

If C is a locally small category, then is there an isomorphism of categories $[C:Sets]^{op} \cong [C^{op}:Sets]$? I feel that there should be since they are both in some sense "spanned" by the re ...
1
vote
0answers
21 views

Reference: Topology on Ind-object

In some articles I've recently seen authors mention that pro-finite groups or pro-finite algebras possess a topology, but they do not explicitly describe it. I was wondering how is the topology ...
2
votes
2answers
28 views

Extending monics in a commutative diagram

Given a commutative diagram in a Grothendieck category $\mathscr{A}$ \begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & ...
4
votes
1answer
52 views

Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
2
votes
0answers
48 views

The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
2
votes
2answers
35 views

Direct product commutes with coproducts?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
0
votes
1answer
41 views

Coproducts and direct products

Is the arbitrary direct sum of modules a submodule of their coproduct? Ie is $\underset{i \in I}{\coprod} M_i \cong \underset{i \in I}{\bigoplus} M_i$... if not then if each $M_i$ where to be ...
3
votes
2answers
60 views

Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and ...
1
vote
1answer
27 views

Cotensor and counit?

If M is a C-bicomodule, then considering C as a $C$-bicomodule also, is $M \square_C C \cong C$, where $\square_C$ is the cotensor product in $^C\mathscr{M}^C$.
0
votes
1answer
15 views

Commutativity with cotensor

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\square_C B \cong B \square_C A$? If not what other conditions are ...
3
votes
0answers
92 views

Category theoretic definition of a “rank” of a subset of an algebraic structure

I am trying to find category theoretic definitions for different possible notions of a rank of a subset (or a subfamily) of an algebraic structure, and to figure out if any of them would be in some ...
4
votes
1answer
58 views

Category example question (Hungerford)

Let $F$ be a free object on the set $X$ (with $i:X\rightarrow F$) in a concrete category $\mathcal{C}$. Define a new category $\mathcal{D}$ as follows. The objects of $\mathcal{D}$ are all maps of ...
1
vote
1answer
38 views

Are cofree comodules Quasi-finite?

Are cofree comodule quasi-finite, where by quasi-finite I mean there is a left adjoint to the cotensor functor?
3
votes
1answer
101 views

Does the naive definition of “commutative category” have any interesting consequences?

By a commutative monoid, let us mean a monoid $A$ in which $a,b \in A$ implies $ab=ba$. Its not at all obvious how to generalize this to the case of an arbitrary category; we cannot just assume that ...
5
votes
1answer
68 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
2
votes
1answer
51 views

Duals of a finite dimensional eveloping coalgebra

Let $C^e$ be the enveloping $k$-coalgebra of a $k$-coalgebra $C$ and denote by ${C^e}^{\star}:=\mathrm{Hom}\,_{k}(C^e,k)$. Then is ${C^e}^{\star} \cong {C^{\star}}^e$?
2
votes
2answers
83 views

Do two definitions for kernels match?

Suppose that we work in Ab, the category of abelian groups. Consider a map $f : A \rightarrow B$ and let $\ker(f) = \{a \in A : f(a) = 0\}$. Now suppose that one can find a map $k : K \rightarrow A$ ...
7
votes
1answer
80 views

Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
12
votes
5answers
637 views

Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
1
vote
1answer
36 views

Adjoint functors and exactness.

If $F, G$ are adjoint functors between two abelian categories, then if $G$ is exact would that imply $F$ is also? If not what assumptions need be made on $F$ or $G$ for this to hold?
10
votes
1answer
99 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
3
votes
2answers
59 views

Does first isomorphism theorem hold in the category of normed linear spaces?

Consider the category of normed linear spaces over $\mathbb{C}$ with bounded linear maps as morphisms. If $M\subset X$ is a subspace, then the quotient space $X/M$ has a map $\|x+M\|: = \inf_{y\in ...
7
votes
2answers
116 views

Categorical viewpoint of $k[x]^\ast\cong k[[x]]$

An exam question which I have found on the internet asks for a "categorical" explanation of $k[x]^\ast\cong k[[x]]$. Could someone help me here? Maybe we have some functorial isomorphism but I don't ...
10
votes
0answers
122 views

Cogroup structures on the profinite completion of the integers

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = ...
7
votes
3answers
305 views

Theorems implied by Yoneda's lemma?

Ok, so I was reading the Wikipedia article on Yonedas lemma. And I've heard before that when you prove things in category theory you automatically get a lot of results by proving it in abstract ...