Functorial properties of the sheaves of higher homotopy groups

$X_1$, $X_2$ irtibatlı, lokal eğrisel irtibatlı topolojik uzaylar,$H_{n_1}$ ve -$H_{n_2}$, de sırasıyla $X_1$ ve $X_2$, üzerinde Yüksek Homotopi Gruplarının Demetleri olsunlar. Eğer $(X_1,H_{n_1})$ ve $(X_2,H_{n_2})$ çiftleri arasında bir F = (f,f*) izomorfizmi varsa, bu taktirde $W_1,subset X_1$, herhangi bir açık cümle, $Gamma(W_1, H_{n_1})$ ve $Gamma(f(W_1), H_{n_2})$de, sırasıyla, $W_1$, ve $f(W_1)$ üzerinde $H_{n_1}$ ve $H_{n_2}$ nin kesitlerinin grupları olmak üzere $Gamma(W_1, H_{n_1})$, $Gamma(f(W_1), H_{n_2})$ ye izomorftur. Üstelik, çiftler ve izomorfizmleri kategorisinden gruplar ve izomorfizmleri kategorisine bir kovaryant funktor vardır

Yüksek homotopi gruplarının demetlerinin funktoryal özellikleri

Let $X_1$, $X_2$ be connected and locally path connected topological spaces, $H_{n_1}$ and $H_{n_2}$ be the sheaves of higher homotopy groups over $X_1$, $X_2$, respectively. If F = (f,f*) is an isomorphism between the pairs $(X_1,H_{n_1})$ and $(X_2,H_{n_2})$ then $Gamma(W_1, H_{n_1})$ is isomorphic to $Gamma(f(W_1), H_{n_2})$ for any $W_1,subset X_1$, open set, where $Gamma(W_1, H_{n_1})$ and $Gamma(f(W_1), H_{n_2})$ are the groups of sections of $H_{n_1}$ and $H_{n_2}$, over $W_1$, and $f(W_1)$, respectively. Moreover, there is a covariant functor from the category of pairs and isomorphisms to the category of groups and isomorphisms.

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