Tagged Questions
3
votes
1answer
67 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 ...
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 ...
10
votes
1answer
388 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 ...
2
votes
3answers
153 views
can the statement “a simplicial set is the nerve of a category if and only if it satisfies a horn-filling condition” be tweaked for groupoids?
For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no higher homotopy groups.
Is ...
3
votes
1answer
236 views
covering spaces and the fundamental groupoid
Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between ...
2
votes
1answer
215 views
Topological Space as an $(\infty,0)$-category
Given a topological space $X$, we may wish to consider it as an $(\infty,0)$-category, where the objects are the points of the space, the 1-morphisms are continuous paths between points, the ...
5
votes
4answers
496 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 ...
15
votes
4answers
286 views
What are the ramifications of the fact that the first homotopy group can be non-commutative, whilst the higher homotopy groups can't be?
Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to ...
