Timeline for Does $\mathsf{Ind}_\lambda^\mu(\mathsf{Ind}_\kappa^\lambda(\mathcal C)) = \mathsf{Ind}_\kappa^\mu(\mathcal C)$?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 29, 2021 at 21:06 | history | bounty ended | Tim Campion | ||
| Jan 27, 2021 at 12:17 | comment | added | Zhen Lin | The matter in differential geometry is confusing. My way of thinking about it: a "real" vector field is neither covariant nor contravariant but invariant. Coordinate basis vectors are covariant, so they are subscripted: $\partial_i x$. Coordinate basis covectors are contravariant, so they are superscripted: $\mathrm{d} x^i$. The components of a vector wrt the coordinate basis are contravariant, so they are superscripted. The components of a covector wrt the coordinate basis are covariant, so they are subscripted. | |
| Jan 27, 2021 at 1:38 | comment | added | Tim Campion | As I'm rambling on anyway, maybe I'll add that although in differential geometry the subscripts are called "covariant" tensors and the superscripts "contravariant", the terminology has always seemed backwards to me, since differential forms are subscripted and are contravariantly functorial in smooth maps, while vector fields are superscripted and covariantly functorial in smooth maps. The real examples of contravariant functors getting superscripts are things like exponentials and duals. | |
| Jan 27, 2021 at 1:32 | comment | added | Tim Campion | In this case, the mnemonic for me was that $\lambda$ is bigger than $\kappa$ and it's conventional to view bigger ordinals / cardinals as lying "above" smaller ones, so I put the bigger one as a superscript. It also meant the subscript ended up in the same place as the usual usage for $Ind_\kappa$. I hadn't thought of it, but you're right -- the variance does also seem super-relevant to making a good choice of notation here. For one thing, it's a good reason not to be using a double superscript or a double subscript. | |
| Jan 27, 2021 at 1:26 | comment | added | Tim Campion | Thanks again! When I went to look up the result of Makkai and Pare you use for Prop 3.5 (their Prop 2.3.11), my copy of their book immediately fell open to that proposition! So perhaps I should have been able to work this out :) But even being able to guess the right answer was probably due anyway to half-remembered conversations with you on this topic from years ago. The point about variance is well-taken -- although the opposite convention is sometimes used, e.g. by Riehl and Verity in Theory and Practice of Reedy Categories (see first line of p. 5). | |
| Jan 27, 2021 at 1:21 | vote | accept | Tim Campion | ||
| Jan 27, 2021 at 0:43 | history | answered | Zhen Lin | CC BY-SA 4.0 |