Tagged Questions

10
votes
1answer
104 views

Functoriality of the Fundamental group

The fundamental group is a functor from the category of pointed topological spaces to the category of groups. Therefore every base-point preserving continuous function $f$ between pointed ...
0
votes
0answers
61 views

A locally constant sheaf on a locally connected space is a covering space; Proof?

As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
2
votes
1answer
86 views

Quotient space and Retractions

I'm trying to learn something about topology and category theory. Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia) For an ...
0
votes
1answer
33 views

Paths between 0-cells in a classifying space. II

Let $\mathcal{C}$ be a small category and $X,Y$ objects within. If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and ...
1
vote
1answer
37 views

Paths between 0-cells in a classifying space.

Let $\mathcal{C}$ be a small category. Giving a morphism $u: X \to Y$ in $\mathcal{C}$ is equivalent to giving a functor $f: \{0 < 1 \} \to \mathcal{C}$ (with $f(0) = X$, $f(1) = Y$, and $f(0 \to ...
3
votes
1answer
68 views

Properties of the Category of topological spaces with $n$ basepoints.

I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to ...
4
votes
2answers
77 views

Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
2
votes
1answer
91 views

Confusion about commutative differential algebra

Here we read the following Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a ...
2
votes
1answer
77 views

Deligne tensor product

I would like to know something about a tensor product of categories and it seems Deligne tensor product is what I am looking for. But the paper "Categories tannakiennes" by Deligne is not available ...
1
vote
0answers
58 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
3
votes
1answer
123 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
1
vote
1answer
68 views

A construction of a pushforward of Vect-presheves.

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
10
votes
1answer
391 views

Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

May's A Concise Course in Algebraic Topology gives the following statement of the Seifert-van Kampen theorem for fundamental groupoids $\Pi(X)$ (section 2.7): Theorem (van Kampen). Let ...
1
vote
0answers
58 views

Can one define “simplicial” EM spaces?

Let $\Delta$ be the well-known simplicial category, and denote with $\widehat\Delta_\bf C$ the category of simplicial objects in $\bf C$ ($\widehat\Delta$ for short). Given $\mathcal ...
10
votes
3answers
392 views

Category Theory usage in Algebraic Topology

First my question: How much category theory should someone studying algebraic topology generally know? Motivation: I am taking my first graduate course in algebraic topology next semester, and, ...
1
vote
1answer
98 views

Left adjoint to inclusion of 1-connected spaces

Is there a left adjoint to the inclusion of the full subcategory of 1-connected spaces into the category of all spaces?
1
vote
1answer
166 views

Meaning of commutative diagram

What is the meaning of a commutative diagram in mathematics? For example, if a map translate an object, then rotate it around the origin and then translate it again, is this a commutative ...
1
vote
0answers
69 views

The Two Eilenberg-Moores

So, there is the Eilenberg-Moore spectral sequence, and there is (for any monad $(T,\mu,\eta)$ on a category $C$) the Eilenberg-Moore Category $C^T$ of $T$-algebras. The silly question, is the ...
3
votes
1answer
179 views

Can someone explain induced homomorphism to me in the context of simplicial homology?

I have two simplicial complexes A and B, and A is a subcomplex of B. We know that there is an inclusion map from A to B, and I understand how to get the simplicial homology groups of each individual ...
1
vote
0answers
147 views

Understanding current development of Category Theory to describe graphs over Euclidean Spaces

This question is about two Mathematical "concepts" joined together in one structure and giving birth to an entity having many descriptions in both domains. Basically I have to deal with graphs. Those ...
0
votes
1answer
140 views

In search of proof that $\widetilde{e^{ix}}:\mathbb{R}\to S^1$ is not epic in $\mathbf{hTop}$

I came across this assertion: There is an epimorphism $X \overset{f}\to Y\;$ in Top such that the homotopy class $X \overset{\tilde{f}}\to Y\;$ of $f$ is not an epimorphism in hTop. Then, by ...
7
votes
5answers
450 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
1
vote
1answer
62 views

Decomposition of simplicial G-sets as a colimits of its simplicial G-subsets

Every simplicial set is the colimit of its finite simplicial subsets. Suppose $G$ be a finite discrete set. Is every simplicial $G$-set a colimit of its finite simplicial $G$-subsets? I'm ...
1
vote
1answer
104 views

Adjointness of the corresponding simplicial functors associated to an adjoint pair between categories

Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. ...
5
votes
1answer
148 views

Why does Frank Adams demand a finite CW-complex?

On page 145 of J.F. Adams' "Stable Homotopy and Generalised Homology", there is a proposition: Let $E$ be the suspension spectrum of a finite CW-complex $K$, and $F$ and spectrum (of topological ...
10
votes
2answers
285 views

Homotopy pushouts and induced maps

Suppose we are in a proper closed model category and consider a commutative square $$ \begin{array}{rcl} A&\to& B\\ \downarrow&&\downarrow\\ C&\to&D \end{array} $$ in its ...
10
votes
2answers
307 views

Introductory book for homotopical algebra

I'm interested in learning homotopical algebra (by which I mean: model categories, simplicial methods, etc.) However, I've been unable to make heads or tails of any of the "standards" ...
5
votes
4answers
498 views

definition of a groupoid

Notation: Underlining $\underline{G}$ denotes a category and $\underline{G}(x,y)$ the class of morphisms from $x$ to $y$. On the Wiki page about groupoids, it is written (I write here my own more ...
1
vote
0answers
78 views

Inverse homology problem

What is the image of the "singular homology" functor $\mbox{Top} \to \mbox{AbComp}$, which associates to each topological space its singular homology chain complex? In other words, is there a (simple ...
5
votes
3answers
347 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 ...
11
votes
3answers
997 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
57 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 ...
3
votes
4answers
290 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 ...
7
votes
2answers
527 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 ...
8
votes
1answer
241 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
11
votes
2answers
296 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. ...