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Stream: learning: questions

Topic: Limit of the identity functor is initial

Thomas Read (Apr 03 2020 at 14:58):

There's a well-known fact that an initial object of a category is the same thing as a limit of the identity functor. The proof is straightforward but (in my opinion) surprisingly fiddly. Is there a "fancy" way to see that this is true / see why this is true?

Sam Tenka (Apr 03 2020 at 19:11):

15859410386471800956770595191858.jpg

Sam Tenka (Apr 03 2020 at 19:13):

From a kan extensions viewpoint, I think the key property is that every functor from 1 to C has an associated cyan triangle through 0-->C and an associated magenta triangle through C-->C

Sam Tenka (Apr 03 2020 at 19:14):

In particular, we may apply the colimit's universal property to the limit, considered as a test arrow. And vice versa

Thomas Read (Apr 03 2020 at 20:33):

I like the diagram! Can you explain a bit more what you mean? Doesn't the magenta triangle correspond to a cone over the identity functor with summit the object corresponding to the functor $1 \to C$, so it can't always exist (e.g. requires the object corresponding to $1 \to C$ to be at least weakly initial)?

John Baez (Apr 03 2020 at 20:43):

I got @Prakash Panangaden into this community today using this URL, which has always been the URL to use if you want to register to join.

Sam Tenka (Apr 03 2020 at 20:58):

@Thomas Read Thanks! The magenta triangle does correspond to that cone over the identity functor, and it indeed does not always exist, for the same reason that limits don't always exist! Observe that a limit needs all "projection maps" and thus will always have arrows to the image of the diagram; so I think your observation that it needs to be weakly initial is true and is contained in the concept of a limit of any diagram that is surjective on objects .

Sam Tenka (Apr 03 2020 at 21:00):

@Thomas Read that said, the diagram I drew is misleading! I think it tells only the truth but that it fails to tell the whole truth. In particular, it sneakily doesn't address the appropriate notion of inverse to show the equality you asked about. So it makes the situation seem simpler than it actually is! I'll try to draw a better diagram tomorrow...

Sam Tenka (Apr 04 2020 at 21:52):

15860371412727975373937176216781.jpg
I didn't figure out a simpler way to draw this. The "proof" here (composition of 2 cells) wraps around a sphere, so hard to draw :/

John Baez (Apr 04 2020 at 22:10):

Some basic stuff that maybe everyone here knows:

A poset is a very simple sort of category. In a poset, including more arrows in a diagram doesn't change its limit. So, we might as well just talk about the limit of a set of objects without bothering to include any arrows in the diagram. The limit, if it exists, is just the greatest lower bound.

The greatest lower bound of the whole poset, if it exists, is the bottom element of that poset: that is, the element that's $\le$ all others.

But the bottom element, if it exists, is also the initial object.

So a special case of "the limit of the identity functor is initial" is "the greatest lower bound of the whole poset, if it exists, is the bottom element".

It's even more fun to think about the greatest lower bound of the empty subset of the poset.

Sam Tenka (Apr 04 2020 at 23:46):

I like this! Yeah, a limit is a greatest lower bound --- "lower" due to the projection maps, and "greatest" due to the universal property. (Just adding on trivially to what you're saying for the sake of obvious obviousness)

Soichiro Fujii (Apr 06 2020 at 12:53):

A generalization of the fact that the limit of the identity functor is an initial object, can be found in Exercise 2.17.2 of Volume 1 of Handbook of Categorical Algebra by Borceux. The following is the dual statement of that exercise, which suits better with the current situation.

Let $F\colon \mathcal{D}\longrightarrow\mathcal{C}$ be a functor and define the category of cocones on $F$, $\mathbf{Cocone}(F)$, as

• an object is a cocone $(M, (r_D\colon FD\longrightarrow M)_{D\in\mathcal{D}})$ on $F$,
• a morphism $(M,r)\longrightarrow (N,s)$ is a morphism $f\colon M\longrightarrow N$ in $\mathcal{C}$ such that $f\circ r_D=s_D$ for every $D\in\mathcal{D}$.

Then $F$ has a colimit if and only if the obvious forgetful functor $U\colon \mathbf{Cocone}(F)\longrightarrow \mathcal{C}$ has a limit. Moreover, if $\mathrm{colim}\ F$ and $\mathrm{lim}\ U$ exist, then they are isomorphic.

Letting $\mathcal{D}$ be the empty category (and $F$ be the unique functor out of it), we obtain the original statement.

A well-known special case is, as John Baez said, the case of posets: a standard proof that a poset $P$ with all infima also has all suprema involves the formula $\bigvee F=\bigwedge (\text{upper bounds of }F)$ (now $F\subseteq P$).