Return to Revisions

2 of 3

Made the question more precise by making the dependency on the objects of $\mathcal{D}$ explicit.

edited Jun 25, 2017 at 14:54

6.1k
1
27
47

Technical term for representing object of a presheaf determined by a left-adjoint?

If $\mathcal{D}$ is a locally-small category, then a functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ has a right-adjoint if and only if for each object $d$ of $D$, the presheaf $\mathcal{C}^{\mathrm{op}}\overset{\qquad c\quad\mapsto\quad \mathcal{D}(F(c),d)\qquad }{\longrightarrow}\mathsf{Set}$ is representable.

Is there a usual technical term for, given $d$, a representing object of the aforementioned presheaf? Or rather, usual or not, have you ever encountered a specialized term for this, and which do you recommend?

_{Remarks. Whether any specialized term is needed, is debatable of course, and not the question. There are situations where it is useful to have a term for it, to facilitate discussing the issue, in particular in expositions. Saying ``representing object of $F$'' would be nonsensical. Of course, one can just describe it the way I did, but is there a specialized technical term for this representing object?
For want of a standard term, and despite the connotations of "law" and similarity to another, usual notation, I always used to call and denote "the" family
$\mathrm{Ob}(\mathcal{D})\ni d\mapsto $(a representing object of the relevant presheaf) $\in$ $\mathrm{Ob}(\mathcal{C})$
"the" $\mathrm{Law}(F)$, for "left-adjoint witnesses of $F$", since if a family $\mathrm{Law}(F)$ exists, then $F$ is a left-adjoint, and such a family of objects is something of a certificate for its being a left-adjoint.

Do you agree that, strictly speaking, it would not be in accordance with the definition of "the" in the to refer to "the law of $F$"?

_{After all, each "component" of $\mathrm{Law}(F)$ is determined only up to isomorphism in the category $\mathcal{D}$, and the feature of any two laws being determined up to _unique_ isomorphism seems to be totally lacking.}}

asked Jun 25, 2017 at 14:26

Peter Heinig

6.1k
1
27
47