The tag has no wiki summary.

learn more… | top users | synonyms

12
votes
4answers
844 views

Degeneracies for semi-simplicial Kan complexes

By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. ...
16
votes
1answer
1k views

Analogue of simplicial sets

This question is prompted by this one (and some of the comments that it drew). Simplicial complex is to ordered simplicial complex as $X$ is to simplicial set. The question is about $X$. Let ...
17
votes
2answers
741 views

Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...
7
votes
2answers
552 views

Historical Question about Simplicial Sets

I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial ...
8
votes
1answer
369 views

delooping under Dold-Kan and simplicial delooping

What maps of simplicial sets exist between the image under the Dold-Kan correspondence of a chain complex shifted up in degree and the image under the right adjoint to simplicial looping of the ...
4
votes
1answer
243 views

Artin-Mazur codiagonal preserves Kan objects?

The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum ...
3
votes
2answers
499 views

Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question. Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...
8
votes
1answer
462 views

Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

If you take the nerve of a groupoid, you get a Kan complex. Question: Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are ...
3
votes
0answers
136 views

Is the homotopy of a primitively generated Hopf algebra still primitively generated?

Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$. Question: is the graded Hopf algebra ...
2
votes
2answers
339 views

Algorithm that decreases the size of the simplicial triangulation

Hello! Let X be a topological space. We are considering only abstract simplicial complexes, i.e. a finite list of finite lists of integers. Is there any algorithm (more or less efficient?), that ...
8
votes
1answer
383 views

Does this “flipping lexicographic” ordering have a standard name?

I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the flipping lexicographic ordering, for evident ...
8
votes
0answers
300 views

A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...
3
votes
2answers
354 views

Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for ...
11
votes
7answers
2k views

History of classifying spaces

Where did the idea and formal definition of the "classifying space of a (small) category" first appear? Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...
4
votes
0answers
58 views

maps of oo-stacks inducing monomorphisms on all homotopy sheaves

What is known about factoring morphisms of simplicial (pre)sheaves/$\infty$-sheaves through maps that induce monomorphisms on all homotopy sheaves? Is there any useful theory for that?
5
votes
1answer
983 views

Geometric Realization of a Simplicial Category

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, ...
3
votes
1answer
192 views

Classifying space of variant on category of simplices

This question might have an easy answer, but my research is far from the region of topology that makes use of classifying spaces of categories, so I can't find it. For (possibly infinite) integers $0 ...
4
votes
4answers
2k views

How to triangulate real projective spaces (as simplicial complexes in Mathematica)?

Hello! I have written a program in Mathematica 7, which calculates for a (finite abstract) simplicial complex all its homology groups. I would really like to test it on the projective spaces, but ...
4
votes
0answers
179 views

What's the best way to test if a sphere is a polytope? (algorithms for the Simplicial Steinitz Problem)

The problem of recognizing whether a simplicial face lattice is polytopal is sometimes called the Steinitz problem. Sturmfels and Bokowski advanced a set of methods in the late 80s to test whether ...
6
votes
0answers
454 views

Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...
2
votes
1answer
229 views

Local injective model structure for simplicial presheaves

The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) ...
4
votes
1answer
230 views

Compatibility of classifying space with inner-hom?

Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category $\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...
6
votes
1answer
784 views

Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...
3
votes
2answers
510 views

From chain complex to simplicial abelian group

In many places, I have seen the slogan that "simplicial abelian group = chain complexes of abelian groups". These same sources usually tell me how to go in one direction. Namely, given a simplicial ...
13
votes
0answers
786 views

Hodge star and harmonic simplicial differential forms

Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set? Let me recall some background. Hodge Theory on a Riemannian manifold A ...
4
votes
1answer
643 views

Simplicial “universal extensions”, the hammock localization, and Ext

Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$ A morphism of ...
19
votes
4answers
1k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
13
votes
3answers
629 views

Triangulations of polyhedra

A topologist came to me with this question, but everything I think should work doesn't. How many triangulations are there of a polyhedron with n vertices? By a "triangulation" of a polyhedron P we ...
8
votes
1answer
1k views

Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representability of nilpotent extensions as trivial extensions over a cofibrant replacement

Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form $$\begin{matrix} R&\to &T\\ ...
18
votes
3answers
1k views

What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?

Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear ...
0
votes
1answer
357 views

How does the discrete group act on simplicial set level by level

Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set? For example, if ...
1
vote
1answer
220 views

Reducing straightening over an interval to straightening over a point

Recall: Let $FU_\bullet:Cat\to Cat_\Delta$ be the bar construction assigned to the comonad $FU$ determined by free-forgetful adjunction $F:Quiv\rightleftarrows Cat:U$. The restriction of ...
10
votes
1answer
326 views

Is the model category of Complete Segal Spaces right proper?

Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has ...
4
votes
4answers
779 views

Homotopic quotients of simplicial sets as infinity-groupoids

Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of ...
13
votes
2answers
1k views

Status of Quillen's conjecture on elementary abelian p-groups

These are questions on D. Quillen's 1978 paper Homotopy properties of the poset of nontrivial p-subgroups of a group. Let $G$ be a finite group, $p$ a prime number, $\mathcal S(G)$ the poset of ...
2
votes
1answer
127 views

Computing the image of the unit map of the join/overcategory adjunction for simplicial sets

Definitions: Recall the definition of the join of two simplicial sets. We may regard the functor $-\star Y$ as a functor $i_{Y,-\star Y}:Set_\Delta\to (Y\downarrow Set_\Delta)$ by replacing the ...
15
votes
2answers
1k views

Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ...
6
votes
0answers
381 views

Simplicial Covering Map

In Rezk's paper "A model for the homotopy theory of homotopy theory" numerous references to simplicial covering maps are made. It's first appearance being at the bottom of page 8. Unfortunately no ...
8
votes
2answers
860 views

Example of a CW complex not homeomorphic to the realization of a simplicial set?

I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't ...
5
votes
1answer
342 views

nerves of crossed complexes, group T-complexes and classifying spaces

A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module. There are a couple of ...
9
votes
4answers
929 views

Is there any generalization of the Dold-Kan correspondence?

The Dold-Kan correspondence gives an equivalence between simplicial abelian groups and chain complexes of abelian groups supported on negative degrees. It actually works for any abelian category. I'm ...
5
votes
1answer
229 views

What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...
5
votes
3answers
1k views

Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

Many topologists express a clear preference for working with CW complexes instead of simplicial sets. One of the reasons is that the cellular chain complex of a CW complex is often easier to work ...
3
votes
1answer
560 views

Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories

Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed. Here is the correct statement of (*): For any cofibration $f:A\to B$ and ...
2
votes
2answers
408 views

Definition of and intuition for regular subdivisions of a polytope

I'm doing a research project that involves subdividing a product of simplices. Specifically, I'm looking at theorem 2.4 from this paper: math.sfsu.edu/federico/Articles/tropOMs.pdf which references ...
4
votes
1answer
493 views

Fibrations of Simplicial sets

Hello, Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations ...
1
vote
1answer
494 views

Proof Sketch: The pullback of the inclusion of the 0th vertex into the standard n-simplex by a right fibration is a deformation retract (450 point bounty if answered by 2am EST)

I was not sufficiently clear on my last attempt at asking a similar (but not identical) question. Tom Goodwillie mentioned (in the accepted answer) that the question can be reduced to this one and ...
1
vote
1answer
307 views

Why is the induced map between pullbacks (of inclusions) by a right fibration a deformation retract?

Let $X$ be a simplicial set. Let $X\to \Delta^n$ be a right fibration (has the right lifting property with respect to right horn inclusions), and let $$\Delta^{\{n-i\}}\hookrightarrow ...
11
votes
2answers
2k views

Understand Cech Cohomology

I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space. I really like Dugger's and Isaksen's paper ...
2
votes
1answer
300 views

Closure of the homotopy relation for a simplicial set

Define a reflexive relation on the set of zero-simplices of a simplicial set $A$ by saying that $x\sim y$ iff there is a one-simplex $h$ with $0$-face $y$ and $1$-face $x$. This is not an equivalence ...