Is there a characterization of right fibrations (by which I mean the invariant $\infty$-categorical notion) along the lines of "conservative, creates colimits + ???"?
The reason I think this could exist is that "on the other side", that is, final functors we have 2 different definitions. One from Quillen theorem which talks about the fibers of the functor and the usual one which is about limits of postcomposition with an arbitrary functor.
@SaalHardali I don't think a right fibration necessarily creates colimits: two points mapping to one point is a right fibration, and either object of the source is mapped to an initial object in the base...
@RuneHaugseng Oh right, of course. I guess I don't know any general property of left/right fibrations except "conservative" (i'm exaggerating of course sorry...). Maybe a more open ended question would be if there's anything like Quillen theorem A for left/right fibrations.
1 hour later…
user131753
5:39 PM
Are questions on homology theory welcome in this room?
@CharlesRezk In our simplicial homology class our professor proved the following theorem,
user131753
"Let $K$ be a finite simplicial complex. If $|K|$ has $k$ many path components then show that the $0$-th homology group of $K$, i.e., $H_0(K)$ has as basis $\{v_i+\partial(C_1(K))\}$ where $v_i\in \operatorname{vert}(K)$. Moreover $H_0(K)$ is isomorphic to $\mathbb{Z}^k$."
user131753
I am looking for a purely algebraic proof of this theorem.
@SaalHardali Isn't Quillen's theorem B basically a criterion to identify the $f\colon C\to D$ which are equivalent to left fibrations whose straightening has the form $D\to \mathcal{S}^{\mathrm{core}} \subseteq \mathcal{S}$?
user131753
5:48 PM
@CharlesRezk I agree that the term is not well-defined. But in this case you can take it to roughly mean that this proof hardly depends on the geometrical properties of $K$.
user131753
In fact, I was thinking along the following lines,
user131753
$C_0(K)$ is a $\mathbb{Z}$-module and has as basis the set $\{v:v\in \operatorname{vert}(K)\}$ (where $\operatorname{vert}(K)$ denotes the set of all vertices of all simplexes in $K$).
A simplicial complex gives you a purely combinatorial object: a collection of finite subsets of the set $V$ of vertices of $K$, which correspond to the simplices of the simplicial complex. This is all you should need.
user131753
So considering the natural epimorphism $f:C_0(K)\to H_0(K)$ (where $H_0(K):=^{C_0(K)}\Big/_{\operatorname{im}(\partial_1)}$) we can see that $f(\operatorname{vert}(K))$ gives a spanning set. I am having trouble proving linear independence of this set.
You have an explicit description of the boundary space: it is spanned by all $(x_1)-(x_0)$ whenever $x_0$ and $x_1$ are vertices connected by an edge $x$.
user131753
5:59 PM
@CharlesRezk Our professor's proof depends on this. But I want to avoid this.
Actually in an earlier class our professor proved that $C_n(K)$ is isomorphic to $\mathbb{Z}^{K_n}$ where $K_n$ denoted the number of $n$-simplexes in $K$. Using this fact it follows that $C_0(K)$ is isomorphic to $\mathbb{Z}^{K_0}$ where $K_n$ denoted the number of $0$-simplexes in $K$, which is same as $|\operatorname{vert}(K)|$.
So, $H_0(K)$ is isomorphic to $^{\mathbb{Z}^{|\operatorname{vert}(K)|}}\Big/_{\operatorname{im}(\partial_1)}$. I want to have an isomorphism between $\operatorname{im}(\partial_1)$ to $\mathbb{Z}^K$ for some $K$.
@CharlesRezk Probably yes. Though I don't see how to apply this to what I seek (probably because you're hinting at something that flew over my head). I'd like to have at least 1 other equivalent definition for being a right fibration which is at least as non-tautological as Quillen Theorem A for final functors.
could we possibly clean out some of the Dropbox? I've run out of Dropbox space yet again (and I've already done tricks to extend my space several times). There's stuff in "misc" and "other papers" that have nothing to do with homotopy theory or number theory I was hoping we could delete
more importantly: www.math.uni-bonn.de/people/scholze/prisms.pdf
