A continuous map $ f : X\to Y$ is a strong shape equivalence if and only if it induces a natural family of equivalences of track groupoids $f_P: \pi P^Y \to \pi P^X$, for all absolute neighborhood retracts P. This suggests that one could define the concept of strong shape equivalence in the more abstract setting of a pair $(C,K)$ of g.e. categories (that is, enriched over the category $GPD$ of groupoids) or even for a 2-functor $F :K\to C$, almost in the same spirit of what happened for ordinary shape equivalences, without considering here coherence conditions. This point of view leads us to the definition of a category $Σ(C,K)$ and of a functor $σ : C →Σ(C,K)$ which inverts exactly the strong shape equivalences of $C$ with respect to $K$. Every g.e. category C has a germ of homotopy in its own structure that becomes evident when one applies the component functor $Δ:GPD\to SET$. This fact allows one to define, for such a g.e. category C, besides a homotopy category $hC$ also a shape category $S(C,K)$, for every chosen full subcategory $K$ of $C$. In the case $C = T OP$ and $K = ANR$, the g.e. category of absolute neighborhood retracts, it is immediate to see that $hTOP$ is the usual homotopy category of all topological spaces and homotopy classes of continuous maps, while $S(T OP,ANR)$ is isomorphic to the ordinary shape category for the pair (hTOP,hANR). We use here a different notation for, e.g., the 2-category T OP and its underlying category TOP. In the first part of the paper we consider the full image factorization of a functor since almost all relevant categories and functors that appear in shape and strong shape are closely related to such a construction. Moreover, it allows one also to clarify (Proposition 1.2) in a simple way the link between the two approaches to ordinary shape by natural transformations and by morphisms of inverse systems. The second part is devoted to g.e. categories and the explanation of what we call the component functor. Here we give our main definitions. The last section is devoted to shape and strong shape equivalences. Also we consider the relationship between the Lisica–Mardeši´c strong shape category $SSh(TOP)$ of topological spaces and the category $Σ(T OP,ANR)$.
Groupoids and strong shape
STRAMACCIA, Luciano
2005
Abstract
A continuous map $ f : X\to Y$ is a strong shape equivalence if and only if it induces a natural family of equivalences of track groupoids $f_P: \pi P^Y \to \pi P^X$, for all absolute neighborhood retracts P. This suggests that one could define the concept of strong shape equivalence in the more abstract setting of a pair $(C,K)$ of g.e. categories (that is, enriched over the category $GPD$ of groupoids) or even for a 2-functor $F :K\to C$, almost in the same spirit of what happened for ordinary shape equivalences, without considering here coherence conditions. This point of view leads us to the definition of a category $Σ(C,K)$ and of a functor $σ : C →Σ(C,K)$ which inverts exactly the strong shape equivalences of $C$ with respect to $K$. Every g.e. category C has a germ of homotopy in its own structure that becomes evident when one applies the component functor $Δ:GPD\to SET$. This fact allows one to define, for such a g.e. category C, besides a homotopy category $hC$ also a shape category $S(C,K)$, for every chosen full subcategory $K$ of $C$. In the case $C = T OP$ and $K = ANR$, the g.e. category of absolute neighborhood retracts, it is immediate to see that $hTOP$ is the usual homotopy category of all topological spaces and homotopy classes of continuous maps, while $S(T OP,ANR)$ is isomorphic to the ordinary shape category for the pair (hTOP,hANR). We use here a different notation for, e.g., the 2-category T OP and its underlying category TOP. In the first part of the paper we consider the full image factorization of a functor since almost all relevant categories and functors that appear in shape and strong shape are closely related to such a construction. Moreover, it allows one also to clarify (Proposition 1.2) in a simple way the link between the two approaches to ordinary shape by natural transformations and by morphisms of inverse systems. The second part is devoted to g.e. categories and the explanation of what we call the component functor. Here we give our main definitions. The last section is devoted to shape and strong shape equivalences. Also we consider the relationship between the Lisica–Mardeši´c strong shape category $SSh(TOP)$ of topological spaces and the category $Σ(T OP,ANR)$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
