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Stream: learning: questions

Topic: exponential, limits

Vincent L (Mar 23 2021 at 18:52):

Is exponential a limit of a diagram ?

Nathanael Arkor (Mar 23 2021 at 20:55):

No, exponentials and limits are distinct (though they are both examples of right adjoints).

Mike Shulman (Mar 24 2021 at 08:46):

Although in $\rm Set$, the exponential $B^A$ is also the product $\prod_{a\in A} B$ of $A$-many copies of $B$.

Javier Prieto (Mar 24 2021 at 12:40):

Hmm surely there's a categorical angle to $e^x = \lim_{n\to \infty} \sum_{k=0}^n \frac{x^k}{k!}$, no? Perhaps buried in here https://www2.math.upenn.edu/~wilf/DownldGF.html

Nathanael Arkor (Mar 24 2021 at 12:52):

@Javier Prieto: the key phrase here is "combinatorial species" (see this article, for instance).

Javier Prieto (Mar 24 2021 at 14:01):

in categorical language, the functor $E: FB \to Set$ [the one with egf $e^x$] is the terminal functor

Well, that's a limit, although not in the way I was expecting.

Dan Marsden (Mar 24 2021 at 16:39):

More generally, if C is a cartesian closed, it's canonically enriched over itself, and then the exponential is a limit in this enriched sense.

John Baez (Mar 24 2021 at 17:26):

Vincent L said:

Is exponential a limit of a diagram ?

People here are exploring different ideas, but I think the best short answer is no.

It's true that exponentiation by $x$ - that is, the functor sending objects $y$ of some category to the exponentials $y^x$ - is commonly defined as the right adjoint to product by $x$ - that is, the functor sending objects $y$ of that category to $x \times y$.

Limits are also right adjoints, but they are right adjoints of a different kind of functor.

John Baez (Mar 24 2021 at 17:27):

(I'm saying all this because this is the basic stuff that most answers here are quickly zipping past, and there could be people lurking here who want to learn the basic stuff.)

Dan Marsden (Mar 24 2021 at 19:20):

Is your exponent in the right place here?

John Baez (Mar 24 2021 at 19:22):

No, "exponentiation by x" should be $y^x$. 'Twill fix.