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Homotopy limits and the homotopy type of functor categories
| Content Provider | Semantic Scholar |
|---|---|
| Author | Cox, David A. |
| Copyright Year | 1976 |
| Abstract | Let Y: / -* Simplicial Sets be a functor. We give a sufficient condition for the map hojim Y —» lim Y to be a weak equivalence. Then we apply this to determine the Artin-Mazur homotopy type of the functor category Funct(7, Sets). 1. Homotopy direct limits. Let / be a small category, and let Y: / -» S be a functor (§ is the category of simplicial sets). In [3, XII 2.1 and 3.7], Bousfield and Kan define holim Y, the homotopy direct limit of Y (Bousfield and Kan work with the pointed category S*, but they remark [3, XII 3.7] that everything remains true in the unpointed case). They also construct a natural map holim Y -» lim Y. Proposition 1 below gives a sufficient condition for this map to be a weak equivalence. First, we need some notation. The "underlying space" or "nerve" of a small category / is denoted N(I) (this differs from the notation of [3, XI 2.1]). If / is an object of /, then l\i is the category of all maps / -> j in / (see [3, XI 2.7]), and we have the formula: Cl) #(A09 = II Hom; (/,/), u (i0 *-«L). y ' u S, we get the functors Y„: / —> Sets (for « > 0), which are defined as follows: if i is an object of /, then Y„(/) is just Y(z')„, the «-simplices of Y(/). Proposition 1. Let Y: / —> § be a functor, and assume that each Y„ is a coproduct of representable functors. Then the natural map: (2) holim Y -» lim Y is a weak equivalence. Proof. By assumption, each Y„ can be written as: (3) Yn = U Hom7(/a, ) Received by the editors July 8, 1975. AMS (MOS) subject classifications (1970). Primary 14F35, 55D99, 18G30; Secondary 14F20, 18F10. |
| Starting Page | 55 |
| Ending Page | 58 |
| Page Count | 4 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1976-0407022-1 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1976-058-01/S0002-9939-1976-0407022-1/S0002-9939-1976-0407022-1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1976-0407022-1 |
| Volume Number | 58 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
