• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.163-191
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• 2002

We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.1
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• pp.137-140
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• 2011

It is known [5] that the concepts of $C_k$-spaces and those can be characterized using by the Gottlieb sets and the LS category of spaces as follows; A space X is a $C_k$-space if and only if the Gottlieb set G(Z, X) = [Z, X] for any space Z with cat $Z{\leq}k$. In this paper, we introduce a dual concept of $C_k$-space and obtain a dual result of the above result using the dual Gottlieb set and the dual LS category.