holim

Limits and colimits in \(\infty\)-category

Initial objects in \(\infty\)-category

In the context to simplicial sets, denote by \(Fun(X,Y)\) the morphism space from \(X\) to \(Y\). We say that a pair of maps \(f\) and \(g\), viewed as edges of \(Fun(X,Y)\) are homotopic if they belong to the same connected somponent of \(Fun(X,Y)\). A pair of maps \(f:X\rightarrow Y\) and \(g:y\rightarrow X\) are called homotopy inverse if \(f\circ g\) and \(g\circ f\) are homotopic to the identity morphisms respectively. In this case we say that \(f:X\rightarrow Y\) is a homotopy equivalance. Let \(X\) be a simplicial set, we say that \(X\) is contactible if the projection map \(X\rightarrow\Delta^0\) is a homotopy equivalance.

3.2.4.3

(Theorem) Let \(X\) be a Kan complex. The following conditions are equivalant

  • \(X\) is contractible.
  • \(X\) is connected and then homotopy groups \(\pi_n(X,x)\) vanish for each \(n>0\) and every choice of base point \(x\in X_0\).
  • \(X\) is connected and then homotopy groups \(\pi_n(X,x)\) vanish for each \(n>0\) and some base point \(x\in X_0\).
  • The projection map \(X\rightarrow\Delta^0\) is a trivial Kan fibration (1.1.5.5) of simplicial sets.

Let \(\mathcal{C}\) be an \(\infty\)-category. We say that an object \(Y\in\mathcal{C}\) is initial if for every object \(Z\in\mathcal{C}\), the morphism space \(Hom_\mathcal{C}(Y,Z)\) is a conractible Kan complex, final is \(Hom_\mathcal{C}(Z,Y)\) is a contractible Kan complex.

4.6.7.11

(Corollary) Let \(X\) be a Kan complex and let \(x\in X\) be a vertex. The following conditions are equivalent

  • \(x\) is initial,
  • \(x\) is final,
  • \(X\) is contractible.