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Stream: learning: questions

Topic: basic question

Jan Pax (Dec 24 2024 at 17:58):

How is $[C^{op}, Set]\rightarrow Set$ defined ?

Nathanael Arkor (Dec 24 2024 at 18:34):

You need to give more context than this. There isn't a unique such functor.

Jan Pax (Dec 24 2024 at 19:04):

I mean $Lan_F$ but still I do not know from where to where $F$ leads and then how is this Lan defined.

Jan Pax (Dec 24 2024 at 19:05):

Or here I mean the induced functor from page 1. Are these two sharpening of my question equivalent ?

Nathanael Arkor (Dec 24 2024 at 19:25):

The linked blog post is talking about [[flat functors]]. If you visit the nLab page, you will find the definition: a functor $F \colon C \to \mathrm{Set}$ induces a functor from the presheaf category since $\mathrm{Set}$ is cocomplete (which is indeed defined by left extension).

Jan Pax (Dec 24 2024 at 19:56):

I've seen the definition in nLab page but I do not follow it. Could you explain this to me in simple terms ? I need a definition involving a simple diagram of arrows.

Nathanael Arkor (Dec 24 2024 at 20:36):

image.png
The universal property of the presheaf category induces a functor $\tilde F$ as in the diagram below, which is exactly the functor you're looking for.

Jan Pax (Dec 24 2024 at 20:56):

How can I see that it is unique ?

Jan Pax (Dec 24 2024 at 21:12):

BTW, what is Sh(D) there ?

Nathanael Arkor (Dec 24 2024 at 21:19):

The universal property of the presheaf construction is proven, for instance, in Awodey's Category Theory (Proposition 9.6).

Nathanael Arkor (Dec 24 2024 at 21:19):

$\mathrm{Sh}$ denotes the category of sheaves on a site.

Jan Pax (Dec 25 2024 at 17:09):

in sheaves, why the morphism for restriction goes in the opposite direction $U\subseteq V$ $res:{\cal F}(V)\to {\cal F}(U)$ ?

fosco (Dec 25 2024 at 21:08):

A sheaf is a contravariant functor...

fosco (Dec 25 2024 at 21:10):

Intuitively, an element of $FV$ is a function defined on $V$: you can restrict it to be a function on each sub-open set of V.

Jan Pax (Dec 26 2024 at 17:17):

Yet one more question: what a monoidal (−1)-category is

fosco (Dec 26 2024 at 20:39):

well, a (-1)-category is a truth value, an element of $\{\top,\perp\}$. If you want to keep having a sequence of inclusions $(-1)\text{-Cat} \subseteq 0\text{-Cat} \subseteq 1\text{-Cat} \subseteq \dots$ then you can represent $(-1)\text{-Cat}$ as the subcategory of $Set$ spanned by the empty set and a singleton.

There aren't many ways in which you can equip one of these with a monoid operation...

Jan Pax (Dec 27 2024 at 19:54):

OK. But I do not even know what is a monoidal n-category, could you help ?

John Baez (Dec 27 2024 at 20:49):

Fosco was trying to tell you that there's just one monoidal -1-category, and it's the truth value "true".

Jan Pax (Dec 28 2024 at 17:28):

@John Baez Yes, I know but I do not know the definition of a monoidal n-category itself.

fosco (Dec 28 2024 at 20:27):

we do not know it either

fosco (Dec 28 2024 at 20:27):

(this is not completely true, but also not completely false)

Nathanael Arkor (Dec 28 2024 at 20:28):

A monoidal n-category is a one-object (n + 1)-category.

Jan Pax (Dec 28 2024 at 21:41):

OK. and what is $(n + 1)$-group $G$ ?

Nathanael Arkor (Dec 28 2024 at 22:40):

Does [[n-group]] not answer your question?