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Stream: learning: questions

Topic: Some General Questions about Equalizers and Notation

Jacob Zelko (Feb 29 2024 at 23:08):

Hi folks,

I've been enjoying a read through of "Category Theory" by Awodey. I had some general questions as I have been reading about equalizers and co-equalizers.

1. I was speaking with Evan Patterson about equalizers and trying to build for myself an intuition about what equalizers "are" as when I first encountered them, they seemed to arbitrarily manifest in the course of my study. Evan gave a great analogy (that I may butcher here) in that you can think of them as "filter"'s that filter out elements between objects. Are there any other analogies that one could think of to describe equalizers?
2. I've attached an image illustrating some of the notation that Awodey is using; could anyone help me understand the notation of what $\top !$ means in this context? Additionally what is the statement $\top \circ !: U \xrightarrow{!}{} 1 \xrightarrow{\top}{} 2$ saying -- my thought was that it is describing a unique map from elements of a subset that match the $\top$ condition in the characteristic function (denoted by $1$) and providing a final map to $2$.

image.png

3. Does $\hookrightarrow$ generally refer to an inclusion?

Thanks everyone!

~ jz

Jacob Zelko (Feb 29 2024 at 23:18):

P.S. Some additional context: I haven't yet learned exactly what are limits/colimits yet. I am first making sure I have a firmer foundation with equalizers as I am moving into learning what are kernels/cokernels which lead naturally to discussions about limits.

Eric M Downes (Feb 29 2024 at 23:18):

! in a category with a terminal object 1, usually seems to refer to the unique map from any object X to 1
! : X --> 1

In Set, there's a special property, which is that the terminal object 1 := {*} also acts as a separator, allowing a unique bijection between elements of an inhabited object (set) X and maps from 1 to X. This approach to set theory is due to Lawvere, the "Elementary Theory of the Catgeory of Sets" which you can read more about here (all quite readable IMO)
-- https://arxiv.org/abs/1212.6543 (Leinster, short)
-- https://www.amazon.com/Sets-Mathematics-F-William-Lawvere/dp/0521010608 (Lawvere, text)

Eric M Downes (Feb 29 2024 at 23:22):

And those refs also cover the use of 2={T, ⊥} as a subobject classifier, and subsets of X as isomorphic to functions from X to 2; which seems to be what Awodey is talking about; so while I haven't read Awodey, I expect that stuff will give you a different view on what you're currently working through, if you get stuck. Good luck!

Jacob Zelko (Feb 29 2024 at 23:24):

I was thinking $!$ was referring to a unique map! I didn't want to assume but glad to see the notational consistency here. And you read my mind -- I was just about to look at Leinster's text on category theory but that arxiv post of his looks great. Lawvere's text is nice; I forgot about it but had come across it before. Thanks for some direction giving @Eric M Downes!

David Egolf (Feb 29 2024 at 23:27):

Hi Jacob!

Regarding (1), if $f,g,:A \to B$ are functions, perhaps we can intuitively think of them as carrying out two different observations on the elements of $A$. Then the set $\{x \in A | f(x)=g(x)\}$ is the collection of all the elements of $A$ where $f$ and $g$ gave the same "output" or "measurement" value. It's like where $f$ and $g$ agree.

Regarding (3), I think $\hookrightarrow$ often refers to an inclusion. The book "Algebra: Chapter Zero" (by Aluffi) says on page 11:

Injections are often drawn $\hookrightarrow$; surjections are often drawn $\twoheadrightarrow$.

Jacob Zelko (Feb 29 2024 at 23:33):

@David Egolf ! Hello! Been a little while but good to hear from you!

That's makes sense about 1. And perfect, glad to know that the inclusion arrow really is the notation for an inclusion -- thought that was the case, but notation will be my downfall Thanks!

David Egolf (Feb 29 2024 at 23:40):

On the topic of working towards limits, you might find it interesting to note that these two things are pretty similar:

• picking out some subset of $A$ where $f$ and $g$ agree
• specifying a set $X$ and two functions $o: X \to A$ and $p:X \to B$ so that this diagram (of sets and functions) commutes:

diagram

For the diagram to commute, we need $f \circ o = p$ and $g \circ o = p$. That means that $p(x)$ has to be equal to $(f \circ o)(x)$ and to $(g \circ o)(x)$ for all $x \in X$. For this to be possible, that means we need $f(o(x)) = g(o(x))$ for all $x \in X$. So our choice of $o:X \to A$ basically involves pointing out particular elements of $A$ where $f$ and $g$ agree.

David Egolf (Feb 29 2024 at 23:51):

Regarding (2), I think $\top!: A \to 2$ is the function that acts by $a \mapsto \top$ for all $a \in A$, where $2 = \{\top, \bot\}$. So it's the function from $A$ to $2$ that is "constant at $\top$".

Actually, I think there might be a typo in that screenshot. I think $!$ should be defined as going from $A$ to $1$, since $\top \circ ! = \top!$ goes from $A$ to $2$.

David Egolf (Mar 01 2024 at 00:00):

One last notational note... Assume we have some function $h:1 \to A$ where $1$ is a set with a single element $*$, and $h(*)=a$, with $a \in A$. Then, we can think of this function as a way of specifying the particular element $a$. Consequently, this function will be sometimes be denoted as $a:1 \to A$.

Josselin Poiret (Mar 04 2024 at 10:10):

David Egolf said:

Regarding (3), I think $\hookrightarrow$ often refers to an inclusion. The book "Algebra: Chapter Zero" (by Aluffi) says on page 11:

Injections are often drawn $\hookrightarrow$; surjections are often drawn $\twoheadrightarrow$.

Note that in CT, those notations rather mean resp. monomorphisms and epimorphisms, which in the category $\mathrm{Set}$
end up being injections and surjections.

James Deikun (Mar 04 2024 at 12:37):

Warning that in homotopy theory the double arrowhead means a fibration, which doesn't have to be a surjection or epimorphism. The cofibrations are usually drawn $\rightarrowtail$ though.