15 apr 2015 -- 18:00
Dipartimento di Matematica, Pisa
Seminario dei Baby-Geometri
Abstract.
Simplicial sets GJ consitute the most natural model for the study of the homotopy theory of spaces: every simplicial set $X_\colon \Delta^\text{op}\to \mathbf{Set}$ can be "topologically realized" in a CW-complex $
X
$, turning the combinatorial informations encoded in its degeneracies into "true" topological informations about the disks and spheres of the cellular structure of $
X
$.
It is extremely natural to regard this construction with categorical eyes, as several "exactness" properties of the geometric realization functor come for free once the correspondence $X\mapsto
X
$ is expressed in the right framework: the "realization" of a simplicial set is a particular example of the so-called "nerverealization paradigm" NR in Homotopical Algebra, examples of which are the "realization" of a simplicial abelian group as a positive chain complex DK, the realization of a simplicial set as a category whose arrows correspond to homotopy classes of 1-simplices J, and many others. This categorical reformulation involves the machinery of coend calculus CC, which is the second main topic of this lecture.
Bibliography
CC Loregian, F., This is the (co)end, my only (co)friend, http:/arxiv.orgabs1501.02503{arXiv:1501.02503}.
DK Kan, D., Functors involving c.s.s complexes, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar.,1958), pp. 330-346
GJ Goerss, Paul G., and John F. Jardine. Simplicial Homotopy Theory. Springer Science & Business Media, 2009.
NR W. G. Dwyer and D. M. Kan, Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147--153.
J Joyal, Andras. Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra 175.1 (2002): 207-222.