12
votes

Accepted

### Original reference for categories of presheaves as free cocompletions of small categories

The earliest reference I can find to the universal property of the presheaf construction is Remark 2.29 of Ulmer's Properties of Dense and Relative Adjoint Functors (1968). However, the proof is only ...

- 5,520

10
votes

Accepted

### Yoneda Lemma for internal presheaves

The Yoneda lemma holds in any finitely complete $2$-category $\mathscr{K}$, such as the $2$-category $\text{Cat}(\mathscr{C})$ of internal categories in a finitely complete category $\mathscr{C}$.
...

- 4,895

10
votes

Accepted

### Yoneda lemma for monoidal categories

The Yoneda lemma is a purely formal result that does not require any size assumptions. For any closed symmetric monoidal category $\mathbf{V}$, any $\mathbf{V}$-category $C$, any object $A\in C$, and ...

- 60.2k

9
votes

### Original reference for categories of presheaves as free cocompletions of small categories

This isn't a direct answer to your question, but you might find the answer by looking in here:
Joachim Lambek, Completions of Categories. Springer Lecture Notes in Mathematics 24, 1966.
The ...

- 25.8k

8
votes

### Is $\prod_{X : \mathcal{U}} (X \to X) \cong 1$ consistent with type theory?

$\prod_{X : \mathcal{U}} (X \to X) \cong 1$ is consistent. It follows from parametricity and function extensionality. Usual parametric models also support function extensionality. The simplest one ...

- 199

7
votes

Accepted

### Yoneda map for a composition of a representable functor and an arbitrary functor

This property of the functor $T$ is called being a dense functor, or "densely generating functor". The notion was first introduced by Isbell in the case where $T$ is fully faithful under the ...

- 51k

6
votes

### Higher and lower analogues of Yoneda's lemma

The fact that you have to consider setiods shouldn't be too surprising, since if we're working ($n>0$)-categorically the distinction between sets and setoids isn't invariant under equivalence. If ...

- 411

6
votes

### The Yoneda Lemma for $(\infty,1)$-categories?

I realize it is a bit late to respond to the question... but this is precisely Lemma 5.1.5.2 of HTT (and the proof is very much along the same lines as the accepted one).

Community wiki

5
votes

Accepted

### Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

This seems like too much to hope for at this level of generality. For example take $T$ to be the "(-1)-truncation" functor
$$
T(\varnothing) = \varnothing, \qquad T(M) = * \,\, \mbox{for $M \...

- 24k

5
votes

Accepted

### Does the following characterize local presentability?

The following argument is from Martin Brandenburg's comment to the question linked to by Omar Antolin-Camarena. As you say, by the Yoneda lemma condition #2 holds iff every continuous functor $C^{op} \...

- 109k

5
votes

### "Philosophical" meaning of the Yoneda Lemma

This perspective seems to be absent so far, even though this is a very old question.
To properly credit the source for this idea, the perspective seems to be taken for granted in MacLane and Moerdijk'...

Community wiki

4
votes

Accepted

### Enrichments vs Internal homs

As Mike says, it is indeed the case that a closed monoidal structure gives rise to a self-enrichment of a category. This is an often-used fact.
Here's an example of "wrong-way" self-enrichment: the ...

- 51k

3
votes

Accepted

### Profunctors as a Kleisli bicategory

This beautiful story take place in $\mathsf{Cat}$, the bicategory of locally small categories. There, the construction of small presheaves induces a pseudomonad whose pseudoalgebrabas are cocomplete ...

- 7,508

3
votes

### Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary

A similar notion (at least in spirit) was introduced by Simon Henry in An abstract elementary class non-axiomatizable in $L_{(\infty,\kappa)}$, see Def. 4.2 and later used by Michael Lieberman, Jiří ...

- 7,508

3
votes

### Is Cauchy completion the largest extension with the same free cocompletion?

The answer is positive.
I found a published account with details to be chapter 6 and 7 of Handbook of Categorical Algebra 1 by Francis Borceux.
Thanks to the comments, useful links that summarize how ...

- 4,207

2
votes

### If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?

I don't know the answer to the original question, but I don't think I'd be alone in arguing that the condition "there is an adjunction terminal among those inducing $T$" doesn't "feel&...

- 51k

2
votes

### Profunctors as a Kleisli bicategory

A nice account of this is given in Martin Hyland's Elements of a theory of algebraic theories.

- 121

2
votes

### Enrichments vs Internal homs

I see that this question has an already accepted answer but I think that may be of interest.
There is a notion of category with internal hom with no reference to a monoidal structure, that is the ...

- 3,153

2
votes

Accepted

### generalized elements in monoidal categories

If I'm not wrong, the following is an answer to 2.
For any objects $A,B,C$ of $\mathcal{C}$, we have the following natural map
\begin{align}
\Phi\colon\mathrm{Hom}(A\otimes B,C) &\to \mathrm{Nat}(...

- 405

1
vote

### Higher and lower analogues of Yoneda's lemma

We can try to apply the Yoneda lemma to $(n,k)$-categories for all $n$ and $k$. If we try to do this for $(0,0)$-categories, then we need the $(0,0)$-category of $(-1,-1)$-categories, and I'm not sure ...

- 4,165

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