Tagged Questions
0
votes
1answer
33 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
37 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 ...
2
votes
1answer
65 views
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 := ...
3
votes
0answers
52 views
What is an “absolute, equational pushout”?
I was leafing through Joyal and Tierney's Notes on simplicial homotopy theory.
In the first few lines of the section on the skeleton of a simplicial set they display the simplicial identities as a ...
1
vote
0answers
139 views
Understanding the definition and notation of geometric realization
I had trouble making sense of the definition of geometric realization of a simplicial set. Let $\Delta^n$ be the standard n-simplex defined as the functor $\hom_\Delta(-,$*n*$): \Delta \rightarrow$ ...
3
votes
0answers
66 views
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, ยง7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
2
votes
1answer
39 views
Colimits of cosimplicial rings
The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings
...
1
vote
0answers
58 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 ...
4
votes
2answers
179 views
Passing pullbacks through adjunction
I'm having trouble following the proof of Proposition I.5.2 in Goerss-Jardine (Simplicial Homotopy Theory). After establishing the adjunction $\hat\Delta(X\times K,Y) \simeq \hat\Delta(K,[X,Y])$, ...
1
vote
1answer
62 views
Decomposition of simplicial G-sets as a colimits of its simplicial G-subsets
Every simplicial set is the colimit of its finite simplicial subsets. Suppose $G$ be a finite discrete set.
Is every simplicial $G$-set a colimit of its finite simplicial $G$-subsets? I'm ...
1
vote
1answer
104 views
Adjointness of the corresponding simplicial functors associated to an adjoint pair between categories
Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. ...
8
votes
1answer
241 views
How does hocolim relate to Hom?
In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
