Compactly generated (T,V)-spaces
 
 
Description: 

First studied by Hurewicz, compactly generated spaces arise as a convenient category for homotopy theory and algebraic topology [1]. The full subcategory of Top (the category of topological spaces and continuous functions) formed by the latter spaces is well-known to have nice properties: it is cartesian closed and coreflective in Top [2] (it is actually the coreflective hull in Top of the class of compact and Hausdorff topological spaces).
Following the approach of [3], we carry the concept of compactly generated spaces into other categories that are topological over Set. To do so, we use the setting of (T,V)-spaces and (T,V)-continuous maps [4] which encodes a general notion of compactness and Hausdorff separation. We recall and discuss conditions under which the previous properties - cartesian closedness and coreflectivity - still hold [5]. Centered in the idea of a generating class, our approach enables a discussion of Alexandroff (T,V)-spaces, and examples are discussed. These results are part of ongoing work [6].

 

[1] Rainer M. Vogt. Convenient categories of topological spaces for homotopy theory. Arch. Math. (Basel), 22:545-555, 1971.
[2] Saunders MacLane. Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971.
[3] Martín Escardó, Jimmie Lawson, and Alex Simpson. Comparing Cartesian closed categories of (core) compactly generated spaces. Topology Appl., 143(1-3):105-145, 2004.
[4] Maria Manuel Clementino and Walter Tholen. Metric, topology and multicategory-a common approach. J. Pure Appl. Algebra, 179(1-2):13-47, 2003.
[5] L. D. Nel. Cartesian closed coreflective hulls. Quaestiones Math., 2(1-3):269-283, 1977/78.
[6] Willian Ribeiro, Compactly generated spaces and quasi-spaces in topology, in preparation.

 

Date:  2018-12-18
Start Time:   15:30
Speaker:  Willian Ribeiro (PhD Student, CMUC)
Institution:  CMUC, Univ. Coimbra
Place:  Room 5.5
Research Groups: -Algebra, Logic and Topology
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