3
$\begingroup$

After I thought of it, I saw the Yoneda lemma, which initially seemed to be the same thing,
but a more careful examination convinced me that the Yoneda lemma is significantly different.


Let morphismsto and morphismsfrom be given by

morphismsto($A$,$\mathcal{C}\hspace{.03 in}$) is the class of all morphisms to $A$ in $\mathcal{C}$
and
morphismsfrom($A$,$\mathcal{C}\hspace{.03 in}$) is the class of all morphisms from $A$ in $\mathcal{C}$

for categories $\mathcal{C}$ and objects A of $\mathcal{C}$.


For any category $\mathcal{C}$, one can form the "class-category" $\operatorname{L}\hspace{-0.03 in}\operatorname{R}(\mathcal{C}\hspace{.02 in})$ whose objects are
$\{\hspace{-0.03 in}$morphismsto($A$,$\mathcal{C}\hspace{.03 in}) : A\in \mathcal{C}\} \:$ and whose morphisms from $\: \{\hspace{-0.03 in}$morphismsto($A$,$\mathcal{C}\hspace{.03 in}) : A\in \mathcal{C}\} \:$ to
$\{\hspace{-0.03 in}$morphismsto($B$,$\mathcal{C}\hspace{.03 in}) : B\in \mathcal{C}\} \:$ are the class-functions $\;\;\; g \: \mapsto \: f\hspace{-0.05 in}\circ \hspace{-0.04 in}g \;\;\;$ for morphisms $\hspace{.04 in}f$ from $A$ to $B$.
Unless I'm missing something here, one can then define a faithful functor $\mathcal{F}\hspace{.02 in}$ from $\mathcal{C}$ to $\operatorname{L}\hspace{-0.03 in}\operatorname{R}(\mathcal{C}\hspace{.02 in})$
by $\;\;\; \mathcal{F}\hspace{.02 in}(A) \: = \: \{\hspace{-0.03 in}$morphismsto($A$,$\mathcal{C}\hspace{.03 in}) : A\in \mathcal{C}\} \;\;\;$ and $\;\;\; (\mathcal{F}\hspace{.02 in}(\hspace{.05 in}f : A\to B))(\hspace{.02 in}g) \: = \: f\hspace{-0.05 in}\circ \hspace{-0.04 in}g \;\;\;$, $\;\;\;$ and that
$\mathcal{F}\hspace{.02 in}$ is such that for all morphisms $\hspace{.04 in}f\hspace{-0.03 in}$ in $\mathcal{C}$, $\hspace{.04 in}f$ is a monomorphism if and only if $\hspace{.02 in}\mathcal{F}\hspace{.02 in}(\hspace{.05 in}f\hspace{.03 in})$ is injective.

By analogy with groups and rings and algebras, I was imagining that construction would
be called the left-regular representation. $\:$ However, searching with google does not
turn up any use of the phrase "left-regular" in any context like what I'm talking about.

I am well aware that $\operatorname{L}\hspace{-0.03 in}\operatorname{R}(\mathcal{C}\hspace{.02 in})$ can have objects and morphisms which are proper classes
even if $\mathcal{C}$ is locally small. $\:$ Are there any other problems with my (attempted?) construction?
Does my (attempted?) construction have a name?



If my construction works, then in cases where the classes morphismsto($A$,$\mathcal{C}\hspace{.03 in}$) are not necessarily sets but the classes morphismsfrom($A$,$\mathcal{C}\hspace{.03 in}$) are necessarily sets, one can get set objects by applying the construction to the opposite category and then using this answer, although I haven't worked out whether or not that would also give the "monomorphism if and only if $\hspace{.02 in}\mathcal{F}\hspace{.02 in}(\hspace{.05 in}f\hspace{.03 in})$ is injective" property.

$\endgroup$
  • $\begingroup$ Maybe I'm tired, but your question is very hard to read. $LR(\mathcal{C})$ is probably either the arrow category of $\mathcal{C}$, or the slice category (resp. coslice category) of objects over $A$ (resp. under $A$), both of which are example of comma categories. But I can't be sure because I'm not able to parse your question. Can you rephrase it without the new terminology ("to-$A$", "from-$A$")? What are the objects of $LR(\mathcal{C})$ exactly? The morphisms...? $\endgroup$ – Najib Idrissi Oct 12 '15 at 19:39
  • $\begingroup$ [cont.] Is $A$ fixed? Variable? And I'm 95% sure that what you call the "left regular representation" is simply the Yoneda embedding. But again, I think it would really help if you wrote all this is plain words. Typically size issues are not really a problem when you're considering concrete applications, though some people have worked (and are still working) on making all this actually well-founded. $\endgroup$ – Najib Idrissi Oct 12 '15 at 19:40
  • $\begingroup$ I figured out a way to avoid the new terminology, although it's far from clear that that helps. $\:$ (I could alternatively have replaced the $\operatorname{Range}$ operators with unions.) $\:$ The descriptions I've seen of the Yoneda embedding refer to local smallness, which is irrelevant for what I described. $\:$ (Unless I messed up, the construction of a concrete class-category works for all categories, and local smallness is not a sufficient condition for the construction I described to produce a non-class category.) $\;\;\;\;$ $\endgroup$ – user57159 Oct 12 '15 at 20:18
  • 1
    $\begingroup$ I also find this question hard to read. In particular, I don't understand what you mean by $\text{Range}$. I also don't know what you think this buys you that the Yoneda embedding doesn't. $\endgroup$ – Qiaochu Yuan Oct 12 '15 at 20:47
  • 1
    $\begingroup$ @Ricky: people already do that using the Yoneda embedding. (And this approach to element-chasing doesn't do everything you'd want it to: for example, the Yoneda embedding does not preserve epimorphisms.) $\endgroup$ – Qiaochu Yuan Oct 12 '15 at 21:08
2
$\begingroup$

Freyd and Scedrov call this construction (seen as a functor $\mathcal{C} \to \text{Set}$ as in Qiaochu Yuan's answer) the Cayley representation in their book Categories, Allegories. They use it to prove the completeness theorem:

Every universally quantified elementary sentence in the predicates of category theory true for the category of sets is true for all categories.

This construction first appeared in the Appendix of the Eilenberg-MacLane paper General theory of natural equivalences, where it is noted that it is an analogue of the left regular representation.

$\endgroup$
2
$\begingroup$

Okay, so here's a version of this construction that makes sense to me. Let $C$ be a small category. There is a functor $C \to \text{Set}$ which takes an object $c \in C$ to the disjoint union $\coprod_{d \in C} \text{Hom}(d, c)$ and which takes a morphism $f : c \to c'$ to the induced morphism

$$\coprod_{d \in C} \text{Hom}(d, c) \to \coprod_{d \in C} \text{Hom}(d, c').$$

This is the composition of the Yoneda embedding $C \to [C^{op}, \text{Set}]$ with the coproduct functor $[C^{op}, \text{Set}] \to \text{Set}$. The Yoneda lemma implies that it's faithful, from which it follows that every small category is concretizable.

I don't know a name for this functor. One way in which it's worse than the Yoneda embedding is that it's not full.

$\endgroup$
  • $\begingroup$ Does the Yoneda embedding yield a "class-category" (version of a category in which objects can be proper classes and morphisms can be class-functions) even when $C$ is not locally small? $\;$ $\endgroup$ – user57159 Oct 12 '15 at 21:35
  • $\begingroup$ @Ricky: the objects of a category aren't either sets or proper classes in the first place: they're just themselves members of a class. If $C$ isn't locally small then the Yoneda embedding doesn't take values in $[C^{op}, \text{Set}]$ but maybe it takes values in something bigger; you can make sense of statements like this using the machinery of Grothendieck universes. $\endgroup$ – Qiaochu Yuan Oct 12 '15 at 21:39
  • $\begingroup$ Grothendieck universes require an increase in consistency strength, so it's not clear to me that "small" results proved using Yoneda embedding on not-locally-small categories can also be shown in ZFC (or even Morse-Kelly + Limitation of Size), whereas the alternative transformation can be interpreted in NBG, and thus "small" results proved that way can also be proven in ZFC. $\;\;\;\;$ $\endgroup$ – user57159 Oct 12 '15 at 21:48
  • $\begingroup$ If you want to do anything serious with not-locally-small categories you should not be working in NBG anyway. $\endgroup$ – Zhen Lin Oct 13 '15 at 6:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy