Timeline for Technical term for representing object of a presheaf determined by a left-adjoint?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jun 28, 2017 at 4:52 | vote | accept | Peter Heinig | ||
| Jun 27, 2017 at 18:35 | answer | added | Peter LeFanu Lumsdaine | timeline score: 2 | |
| Jun 27, 2017 at 15:18 | history | edited | Leo Alonso | CC BY-SA 3.0 |
Fixed LaTeX formula
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| Jun 27, 2017 at 12:35 | answer | added | Todd Trimble | timeline score: 4 | |
| Jun 26, 2017 at 17:22 | comment | added | Dylan Wilson | (Or I should say 'Cartesian arrow covering the non-identity arrow in [1]') | |
| Jun 26, 2017 at 17:19 | comment | added | Dylan Wilson | Yeah that wasn't clear: I mean consider the category whose objects are either objects of C or objects of D and morphisms between an object of C and an object of D are Hom(Fc,d). This category has a projection to [1] where you send C to 0 and D to 1. That projection map is a cocartesian fibration. The data of a Cartesian arrow in this category is equivalent to the data of what we've at this point agreed to call a 'universal arrow'. | |
| Jun 26, 2017 at 15:55 | comment | added | Peter Heinig | @ToddTrimble: useful comments, many thanks. I was curious whether in there is a was a specialized term in the context of right-adjoints. But, yes, using the standard terminology of universal arrows seems the most sensible thing to do. Would you mind making the comments an answer so I can check the question as answered? | |
| Jun 26, 2017 at 15:53 | comment | added | Peter Heinig | @DylanWilson: what is meant by "correspondence over the poset [1]"? | |
| Jun 26, 2017 at 4:12 | comment | added | Mike Shulman | Another possibility is a "cofree object": ncatlab.org/nlab/show/free+object | |
| Jun 25, 2017 at 17:40 | comment | added | Todd Trimble | Finally, I'd prefer to say not that the object $c'$ is a "witness", but it's the $\theta$ (in the notation of the preceding comment) that witnesses the representability by $c'$. Thus $\theta \in D(F c', d)$ is a witnessing term or element, similar to other uses of the word in category theory and type theory. | |
| Jun 25, 2017 at 15:16 | comment | added | Todd Trimble | So my answer to your "strictly speaking" is yes, I see the point you're making, but I think of this as really just one of those abuses of language that we all learn to tolerate when, for example, we use a single letter $X$ to denote a measure space which is actually an ordered triple $(X, \Omega, \mu)$. So a "representing object" $c'$ should really be regarded as a shorthand for an ordered pair $(c', \theta: Fc' \to d)$ that is the universal thing and that by all rights is what we mean by "the representing object". | |
| Jun 25, 2017 at 14:56 | comment | added | Todd Trimble | Mac Lane in Categories for the Working Mathematician wisely refers instead to a universal arrow from $F$ to $d$, thus placing emphasis on the fact that it's not just a representing object $c'$ involved but an ordered pair $(c' \in Ob(C), Fc' \to d)$ which is universal, i.e., terminal (as an object in the comma category $F \downarrow d$). This takes care of your other question too since we can speak of "the" universal arrow in the sense you link to. | |
| Jun 25, 2017 at 14:54 | history | edited | Peter Heinig | CC BY-SA 3.0 |
Made the question more precise by making the dependency on the objects of $\mathcal{D}$ explicit.
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| Jun 25, 2017 at 14:40 | comment | added | Dylan Wilson | I don't know of a term. In a given discussion I might call it "a model for the right adjoint evaluated at d". If you re-write the functor F as a correspondence over the poset [1], then you might call such a representing object the 'source of a Cartesian arrow whose target is d'. | |
| Jun 25, 2017 at 14:26 | history | asked | Peter Heinig | CC BY-SA 3.0 |