3
votes
2answers
70 views

How do we get a simplicial homology functor?

The $n$-th simplicial homology group $H_n(A)$ of an abstract simplicial complex $A$ depends on the choice of an orientation for $A$ (but for different orientations, the homology groups are ...
4
votes
2answers
77 views

(Certain) colimit and product in category of topological spaces

Consider the diagram $$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$ in category of topological spaces. Denote $I$ the ...
3
votes
1answer
45 views

How do you construct the lifted topology of a groupoid cover?

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...
6
votes
0answers
63 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
2
votes
0answers
35 views

Notation and shorthand of cobordism G=O, SO, U

This is really a basic question: From Wikipedia: "The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex ...
3
votes
1answer
190 views

Algebraic topology and homotopy in category theory

I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
1
vote
1answer
62 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
2
votes
1answer
58 views

Constructing a ring in Top by wedging and smashing pointed spaces

I'll list some things I believe I can do, any of which might I might not actually be able to, and then, if I'm right, I'll ask if there's any point to what I've done. The category of pointed locally ...
3
votes
0answers
51 views

Can injective and projective model structures be the same ?

Is there a non trivial example where injective and projective model structures coincide ? By non trivial I mean an example of a functor category on a non discrete base category where I guess ...
1
vote
0answers
69 views

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of ...
3
votes
2answers
85 views

How can a functor which preserves products not be natural?

Since the functor $\pi_1$ preserves products, how can $\pi_1(T,t_0)\approx\pi_1(S^1,x_0)\times\pi_1(S^1,y_0)$ be not natural? Here $T$ is the torus. Wikipedia gives an explanation, but I am still a ...
1
vote
0answers
23 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 ...
3
votes
1answer
57 views

Question on the uniqueness of a homotopy colimit up to unique isomorphism

Let me first give an abstract definition of the homotopy colimit. Let $C$ be a cofibrantly generated model category and let $D$ be a small category. There is an adjunction $$ ...
3
votes
1answer
68 views

n-truncated simplicial set

It might be a trivial question. So, I apologise in advance. Let $ \Delta ^{op}_n $ be the full subcategory of $ \Delta ^{op} $ such that the set of objects of $ \Delta ^{op}_n $ is $ \left\{ 0, ...
1
vote
1answer
60 views

1-1 correspondence between nuclei and regular monomorphisms of a locale

I am having a little trouble with Theorem 2.3 in Professor Johnstone's book on "Stone Spaces". The theorem depends on a Lemma (which I am not struggling with; I only include it for context) which ...
5
votes
1answer
104 views

$2$-Morphisms in the Fundamental $2$-Groupoid

I'm trying to write down a clean definition of the fundamental $2$-groupoid $\pi_{\leq 2}(X)$ of a topological space $X$. Specifically, I'm concerned with how to properly define $2$-morphisms. Here is ...
4
votes
1answer
82 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
1
vote
0answers
55 views

Is it possible to consider this property of “being nice” as a homotopy property?

Consider a Model (Quillen) Category $ M $ (possibly with all oobjects being fibrant or cofibrant (or both)). I'm wondering if the following property is a homotopy property: "Let $ p $ be a morphism ...
8
votes
2answers
200 views

Why does the loopspace $\Omega$ induces a weak equivalence on mapping telescopes?

I am trying to answer an exercise of Hatcher's "Algebraic Topology", Section $4$.F, exercise $3$. Suppose we are given a sequence of pointed topological spaces : $Z_0\rightarrow Z_1\rightarrow Z_2 ...
1
vote
1answer
76 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
7
votes
1answer
151 views

What exactly is duality?

In general, I am familiar with this notion of duality (i.e. in category theory, a statement is dualized simply by "reversing all arrows" and leaving objects unchanged). There are a couple of questions ...
2
votes
1answer
48 views

Understanding Quillens Theorem A

Let me restate the theorem: Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. If $F\downarrow x$ is contractible for every $x\in\operatorname{Ob}(\mathcal{D})$, then $F$ is a homotopy ...
4
votes
1answer
78 views

Visualizing a homotopy pull back

I am currently taking a course in algebraic topology, which also covers a lot of category theory. My question is pretty straightforward: How do you visualize the (homotopy) pull back of a diagram ...
1
vote
1answer
71 views

Question on functors

please i need help,how to prove that "the functor (covariant) "fundamental group", of the category of pointed topological spaces in the category of groups" is really a functor What i must do to ...
1
vote
0answers
66 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
2
votes
1answer
73 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
4
votes
2answers
146 views

Can abstract nonsense be helpful here?

Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top ...
1
vote
3answers
107 views

Symmetric monoidal products that preserve limits and colimits

Are there common examples of a symmetric monoidal product $\otimes$ that preserves both limits and colimits in each variable? This question is worded incorrectly, I now realize: (A) I originally ...
1
vote
2answers
75 views

Changing the direction of the homomorphism in the definition of pushouts

In the diagram below $P$ is a pushout of the data if a unique $u:P\rightarrow Q$ exists for every solution $(Q,j_1,j_2)$ (by the definition of bushouts). My question is : Suppose that the definition ...
1
vote
1answer
62 views

Fixing an object $Y$ defines a contravariant functor $\mathcal{C}^{op} \rightarrow$ Sets

Say we fix an object $Y$, and then consider the assignment $(\mathcal{C}^{op})_0 \rightarrow$ Sets that sends $X$ to $\hom_{\mathcal{C}}(X, Y)$. How can I show that this defines a contravariant ...
4
votes
1answer
68 views

When is the functor $\otimes:\mathcal{C}^\omega\times\mathcal{C}^\omega$ is monoidal functor?

Admittedly, the question in the title is not a yet a precise question, so I must make it a precise question. Let $\mathcal{C}$ be a monoidal category, with monoidal product, $\otimes$. We will let ...
5
votes
1answer
150 views

Homotopic maps in a directed system induce homotopic maps on colimit?

Let $(A_i,f_i)$ be a directed system of CW-complexes with colimit $A$. Further, let $g_i:A_i\to A_{i+1}$ be maps such that $g_{i+1}f_i=f_{i+1}g_i$ and $f_i\simeq g_i$. This might or might not be the ...
11
votes
1answer
127 views

The fundamental group functor is not full. Counterexample? Subcategories with full restriction?

Anyone aware of a nice counterexample to "The fundamental group functor is full?" (Which is...false, right?) and are there a nontrivial subcategories on which its restriction is full? I.e. Can you ...
1
vote
0answers
60 views

Direct limit of $CW$ complex and infinite Stiefel manifold

Let $V_{n}(\mathbb{R}^k)$ be the Stiefel manifold of ortogonal $n$-frames in $\mathbb{R}^k$ and $G$ a compact Lie group. A classifyng space for group $G$ is a connected topological space $BG$, ...
10
votes
1answer
179 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 ...
7
votes
2answers
339 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
178 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
36 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
41 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
105 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 ...
5
votes
3answers
206 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
113 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
111 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
62 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
241 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
78 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
643 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
61 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 ...
11
votes
3answers
705 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
102 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?