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Stream: learning: questions

Topic: Is this definition for the slice category correct?

Davi Sales Barreira (May 14 2022 at 16:55):

Hello, friends. I've been reading the great "Category Theory for Scientists" book. Now, I was confused when I got to the definition of the slice category.
image.png

My question is whether the $\alpha \circ i = \text{id}_X$ is correct, or if the correct formulation would be $\alpha \diamond i = \text{id}_X$ .

Matteo Capucci (he/him) (May 14 2022 at 16:57):

What's the difference between $\diamond$ and $\circ$? What's a 'left cone' in the def?

Todd Trimble (May 14 2022 at 17:25):

Davi Sales Barreira said:

Hello, friends. I've been reading the great "Category Theory for Scientists" book. Now, I was confused when I got to the definition of the slice category.
image.png

My question is whether the $\alpha \circ i = \text{id}_X$ is correct, or if the correct formulation would be $\alpha \diamond i = \text{id}_X$ .

Although some of the terminology and notation doesn't seem standard to me, I think the way it's written is correct. (I don't know what $\diamond$ stands for either.)

The left cone undoubtedly means the category obtained by freely adjoining an initial object to the diagram category $I$.

Objects of the slice amount to tuples $(c; \{f_i: c \to X(i)\}_{i \in I})$ where $c$ is an object of $C$ and $f_i$ is a morphism of $C$, one for each object $i$ of $I$, such that for any morphism $g: i \to j$ in $I$, we have $X(g) \circ f_i = f_j$. A morphism of the slice, from $(c; f_i)$ to $(d, g_i)$, amount to morphisms $h: c \to d$ such that $f_i = g_i \circ h$ for all objects $i$ of $I$.

Reid Barton (May 14 2022 at 17:29):

The composition in the last equation is really a whiskering; maybe the book writes that as $\diamond$?

Davi Sales Barreira (May 14 2022 at 18:28):

Thanks for the answers! The left cone would be the original category with an additional additional $Lc$ initial object.
Yeah, the $\diamond$ would be the whiskering. In the book, the $\circ$ is the vertical composition for natural transformations. Thus why I thought there might be a typo. So, if the correct definition is with a whiskering, then I guess it's just a typo indeed.

Morgan Rogers (he/him) (May 15 2022 at 11:09):

Todd Trimble said:

Davi Sales Barreira said:

Hello, friends. I've been reading the great "Category Theory for Scientists" book. Now, I was confused when I got to the definition of the slice category.
image.png

My question is whether the $\alpha \circ i = \text{id}_X$ is correct, or if the correct formulation would be $\alpha \diamond i = \text{id}_X$ .

Although some of the terminology and notation doesn't seem standard to me, I think the way it's written is correct. (I don't know what $\diamond$ stands for either.)

The left cone undoubtedly means the category obtained by freely adjoining an initial object to the diagram category $I$.

Objects of the slice amount to tuples $(c; \{f_i: c \to X(i)\}_{i \in I})$ where $c$ is an object of $C$ and $f_i$ is a morphism of $C$, one for each object $i$ of $I$, such that for any morphism $g: i \to j$ in $I$, we have $X(g) \circ f_i = f_j$. A morphism of the slice, from $(c; f_i)$ to $(d, g_i)$, amount to morphisms $h: c \to d$ such that $f_i = g_i \circ h$ for all objects $i$ of $I$.

I would call this the category of cones over $X$, although it does coincide with the usual definition of slice category when $I$ is the one-object category.

Todd Trimble (May 15 2022 at 11:50):

Morgan, I didn't write the book. Or do you think I transcribed it incorrectly?

Reid Barton (May 15 2022 at 12:05):

FWIW, Lurie also uses the "slice" terminology (https://kerodon.net/tag/017W)

Morgan Rogers (he/him) (May 15 2022 at 13:32):

Todd Trimble said:

Morgan, I didn't write the book. Or do you think I transcribed it incorrectly?

Of course, I just thought it was worth mentioning another name in case @Davi Sales Barreira sees/has seen that elsewhere.

Davi Sales Barreira (May 16 2022 at 11:37):

I would call this the **category of cones over $$X$$**, although it does coincide with the usual definition of slice category when $$I$$ is the one-object category.

@Morgan Rogers (he/him) , they only coincide for $I$ with one-object? That's strange. What is the difference in relation to the "actual" slice category?

Morgan Rogers (he/him) (May 16 2022 at 11:44):

I mean that usually when one talks about a slice category, it is the slice over a particular object; see [[over category]]. It is mentioned there that the slice is a special case of a comma category, but that's a different definition from the category of cones.

James Deikun (May 16 2022 at 12:28):

Comma categories are ultimately a more general construct than categories of cones, when used in conjunction with the rest of the categorical machinery; you can recover the category of cones of $X : J \to C$ as a comma category $(\Delta/X)$ of the functor category $C^J$ (where $\Delta : C \to C^J$ is the diagonal functor and $X$ is used as the constant functor with value $X$ in the usual abuse of notation). I'm not aware of any similar way to recover comma categories with a nontrivial use of a category of cones ...

Davi Sales Barreira (May 16 2022 at 15:18):

Morgan Rogers (he/him) said:

I mean that usually when one talks about a slice category, it is the slice over a particular object; see [[over category]]. It is mentioned there that the slice is a special case of a comma category, but that's a different definition from the category of cones.

I see. Thanks a lot for this. This actually clarified why I saw different definitions out there. Now it makes more sense. Indeed, the first time I saw the slice category, it was the definition you are suggesting.

Let me just check one thing. The limit $\lim_{I} X$ would be the terminal object of what you called the category of cones over $X$. Right?

Davi Sales Barreira (May 16 2022 at 20:25):

Awesome. Thanks!