# Tag Info

Accepted

### Basic Example of Yoneda Lemma?

Computer scientists do use quite a few algebraic structures, they are just slightly different from the ones mathematicians usually focus on. Computer scientists tend to care about monoids, semigroups,...
• 25.2k
Accepted

• 18.7k
Accepted

### Is there a Yoneda lemma for categories other than Set?

There are two ways to interpret your question. Identifying the role of Set in the Yoneda lemma as the category where your categories are enriched in reveals that if you consider $\mathcal V$-enriched ...
• 80.4k
Accepted

### A particular dualization of the Yoneda lemma fails to exist

It's not a size question, as the same size question could be asked for the usual Yoneda lemma. To understand the exercise, it is important to recall what the duality principle tells us in category ...
• 44k
Accepted

### Question on the “Yoneda perspective”

This is against the philosophy of the Yoneda Lemma. When we want to prove $X \cong Y$ via the Yoneda Lemma, the whole idea is to look at the whole category and compare how $X$ and $Y$ relate to its ...
Accepted

• 52.9k
Accepted

### Categories for which every contiuous sheaf is representable

Without the cocompleteness condition, these were studied in G. M. Kelly's paper "A survey of totality for enriched and ordinary categories" under the name compact categories, and in this mathoverflow ...
• 16.4k
Accepted

### Is Yoneda Lemma a characterization of isomorphism?

I think there is nothing subtle here (except for Yoneda lemma, of course). A consequence of Yoneda lemma is that the Yoneda embeddings $X \mapsto C(-, X), \; X \mapsto C(X, -)$ are fully faithful ...
• 8,045

### Why presheaves are generalized objects?

The previous answers are very good, but I also like to always keep in mind a simple example when working with presheaves, to get a feel for what all this means. Luckily, we have a very simple and ...
• 1,272
Accepted

### On functors agreeing with the powerset functor on objects and not being isomorphic to it

There seems to be quite a lot of literature on functors from sets to sets that are determined, up to isomorphism, by their values on objects ("DVO functors"). I found a paper (reference below) which ...
Accepted

### Is the Yoneda completion of the rationals the extended real line?

You're right; this is an error in what you've heard. The embedding $i:P\to \hat P$ of a poset $P$ into its complete poset of downsets $\hat P$ is the free completion of the poset, in the sense that ...
• 52.9k
Accepted

### Exponents in a slice category of presheaves

Let me first treat the case of the topos of sets, and then generalize. In this case, if $F$ is a set and $x : X \to F$ is an object of $\mathbf{Set}/F$, then this object is uniquely determined up to ...
• 21.2k
Accepted

### Confusion about the Yoneda lemma

Here's one possible answer to this question. Let's take the viewpoint that functors are representations of categories. First, why is this sensible? Well, recall that categories are generalizations of ...
• 28.8k