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Stream: deprecated: id my structure

Topic: "Weak" cones

Moana Jubert (Aug 12 2023 at 14:04):

Hi, I was wondering if there is a standard name/concept for the following construction: given a diagram $D : \mathcal{I} \to \mathcal{C}$ and an object $c \in \mathcal{C}$, I define a "weak" cone under $c$ to be a functor $W : \mathcal{I} \to \mathbf{Set}$ such that 1) $W_i \subseteq \mathrm{Hom}_{\mathcal{C}}(c, D_i)$ and 2) morphisms $W_f : W_i \to W_j$ are just postcomposition by $D_f$ : $\phi \in W_i \mapsto D_f \circ \phi \in W_j$.

Then a cone is just a "weak" cone where every $W_i$ is a singleton. Or, conversely, a "weak" cone is a cone where the "arms" of the cone are no longer unique.

I would be very interested if this has been used somewhere else!

Moana Jubert (Aug 12 2023 at 19:50):

Erf I'm sorry I didn't see there's #learning: id my structure entirely dedicated to this kind of questions—better to move the topic there maybe

Morgan Rogers (he/him) (Aug 12 2023 at 19:52):

Moved :+1:

James Deikun (Aug 12 2023 at 20:15):

Let's see ... this is the same as a subfunctor of $\mathrm{Hom}_{\mathcal{C}}(c,D-)$. In $c \downarrow D$ it's a sieve (or is it cosieve?).

Patrick Nicodemus (Aug 13 2023 at 04:55):

Moana Jubert said:

Hi, I was wondering if there is a standard name/concept for the following construction: given a diagram $D : \mathcal{I} \to \mathcal{C}$ and an object $c \in \mathcal{C}$, I define a "weak" cone under $c$ to be a functor $W : \mathcal{I} \to \mathbf{Set}$ such that 1) $W_i \subseteq \mathrm{Hom}_{\mathcal{C}}(c, D_i)$ and 2) morphisms $W_f : W_i \to W_j$ are just postcomposition by $D_f$ : $\phi \in W_i \mapsto D_f \circ \phi \in W_j$.

Then a cone is just a "weak" cone where every $W_i$ is a singleton. Or, conversely, a "weak" cone is a cone where the "arms" of the cone are no longer unique.

I would be very interested if this has been used somewhere else!

this is very close to what is called a "weighted" cone, and the analogous notion of limit is called a weighted limit.

James points out that this is a subfunctor $m: P \to Hom_C(c,D(-)$. In general, for a weighted cone, $P$ need not be a subfunctor, but it can be an arbitrary presheaf, and $m$ can be an arbitrary natural transformation.

Or in other words, instead of $W_i\subset Hom_C(c,D_i)$, we have a map $f_i : W_i \to Hom_C(c,D_i)$, and this reduces to your situation when we have $f_i$ injective.

Patrick Nicodemus (Aug 13 2023 at 05:01):

The notion of weighted limit is essential in enriched category theory, where ordinary limits often cannot be defined.

Moana Jubert (Aug 13 2023 at 07:38):

Of course! This looks promising. Thank you!