Articles of simplicial stuff

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= conv(e_2,e_1\!-\!e_2)$ and $\sigma_1^\vee= conv(e_2\!-\!e_1,e_1)$ and $\tau^\vee= conv(e_2\!-\!e_1,e_1\!-\!e_2,e_1,e_2)= conv(e_2\!-\!e_1,e_1\!-\!e_2,e_1)= conv(e_2\!-\!e_1,e_1\!-\!e_2,e_2)$. The corresponding semigroup algebras are $S_{\sigma_0}= \mathbb{C}[\mathbf{x}^{e_2},\mathbf{x}^{e_1\!-\!e_2}] =\mathbb{C}[y,xy^{-1}]$ and $S_{\sigma_1}= \mathbb{C}[\mathbf{x}^{e_2\!-\!e_1},\mathbf{x}^{e_1}]= \mathbb{C}[x^{-1}y,x]$ and $S_\tau= \mathbb{C}[\mathbf{x}^{e_2\!-\!e_1}, \mathbf{x}^{e_1\!-\!e_2}, \mathbf{x}^{e_1},\mathbf{x}^{e_2}]$ $=$ $\mathbb{C}[x^{-1}y,y^{-1}x,x,y]$ $=$ $\mathbb{C}[x^{-1}y,y^{-1}x,x]= \mathbb{C}[x^{-1}y,y^{-1}x,y]$ inside $\mathbb{C}[x^{\pm1},y^{\pm1}]$. The three associated affine […]

Show simplicial complex is Hausdorff

I have a simplicial complex $K$ and I need to show that its topological realisation $|K|$ is Hausdorff. And $K$ need not be finite. I have very little idea on how to get started on this. Only that if $x,y \in |K|$, I need to find disjoint open sets containing $x$ and $y$. I also […]

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The following are equivalent: $f : X \to Y$ is a homotopy equivalence. $f_* : \pi_0 (X) \to \pi_0 (Y)$ […]

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and morphisms $d_2^0, d_2^1, d_2^2 : C_2 \to C_1$, $d_1^0, d_1^1 : C_1 \to C_0$, $s_0^0 : C_0 \to C_1$, satisfying the following fragment of the simplicial […]

Product of simplicial complexes?

Given two abstract simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, what is the definition of their product $\mathcal{K} \times \mathcal{L}$, as another abstract simplicial complex? Basically I’m looking for the definition of “product” such that $|\mathcal{K} \times \mathcal{L}|$ is homeomorphic to $|\mathcal{K}| \times |\mathcal{L}|$, if that makes sense ($|\cdot|$ denotes geometric realization; not sure if this is […]

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I’ve heard people speaking about combinatorial homotopy theory, but every reference apparently related seems to deal with general concepts of algebraic topology like CW-complexes, homotopy […]

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my definitions are not precisely correct: The category of simplicial sets is a cofibrantly generated model category with generating acyclic cofibrations the inclusions of […]

adjunction relation

Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set $\operatorname{Hom}(X,Y)_n := \operatorname{Hom}_{sSet}(\Delta^n\times X,Y)$ where $\Delta^n := \operatorname{Hom}_{\Delta}(\cdot,[n])$. Then I have the ‘adjunction relation’: $\operatorname{Hom}(Z,\operatorname{Hom}(X,Y)) \simeq \operatorname{Hom}(Z\times X,Y)$ Can I do the same thing for simplicial topological spaces $X,Y:\Delta^{op}\rightarrow \operatorname{Top}$? Are there references on that or is that […]