Intereting Posts

Definition of the Young Symmetrizer
Proof that a trigonometric function of a rational angle must be non-transcendental
Is there a simple math behind splitting multiple bills evenly across many people?
Proving that the limit of a sequence is $> 0$
'Proof' of the correspondence between maximal ideals and points in projective space
graph is dense in $\mathbb{R}^2$
True vs. Provable
Find a minimum of $x^2+y^2$ under the condition $x^3+3xy+y^3=1$
special values of zeta function and L-functions
Original Papers on Singular Homology/Cohomology.
Closed form of a series involving the squared Beta function
Can compact sets completey determine a topology?
Galois group of the splitting field of the polynomial $x^5 – 2$ over $\mathbb Q$
Is it valid to write $1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$?
Is there an analytic function with $f(z)=f(e^{iz})$?

I’ve recently started reading *Categories for the Working Mathematician* and I’m a little puzzled about the distinction between $\textbf{Set}$ and $\textbf{Ens}$. On the one hand, it seems like $\textbf{Ens}$ is supposed to be a full subcategory of $\textbf{Set}$:

If $V$ is any set of sets, we take $\textbf{Ens}_V$ to be the category with objects all sets $X \in V$, arrows

allfunctions $f : X \to Y$, with the usual composition of functions. By $\textbf{Ens}$ we mean any one of these categories.

And on the other hand it seems like $\textbf{Ens}$ can be bigger than $\textbf{Set}$:

- How does $C$ small imply $Set^{C^{op}}$ locally small?
- Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ sending groups to their centers?
- Additive functor over a short split exact sequence.
- Does this category have a name? (Relations as objects and relation between relations as morphisms)
- What's an Isomorphism?
- If the cohomology of two objects in the derived category are equal, are the objects isomorphic?

$D$ must have small hom-sets if these functors are to be defined […]. For larger $D$, the Yoneda lemmas remain valid if $\textbf{Set}$ is replaced by any category $\textbf{Ens}$ […] provided of course that $D$ has hom-sets which are objects in $\textbf{Ens}$.

I suspect my confusion is arising from a misinterpretation of what $\textbf{Set}$ is and what it is used for. Mac Lane seems to make a point of building $\textbf{Set}$ using a fixed universal set $U$, but is there any benefit to doing this instead of simply taking $\textbf{Set}$ to be his “metacategory” of *all* sets? Indeed, set-theoretically, can such a universal set exist? Mac Lane requires the following properties:

- $x \in u \in U$ implies $x \in U$,
- $u \in U$ and $v \in U$ implies $\{ u, v \}$, $\langle u, v \rangle$, and $u \times v \in U$.
- $x \in U$ implies $\mathscr{P} x \in U$ and $\bigcup x \in U$,
- $\omega \in U$ (here $\omega = \{ 0, 1, 2, \ldots \}$ is the set of all finite ordinals),
- if $f : a \to b$ is a surjective function with $a \in U$ and $b \subset U$, then $b \in U$.

As far as I can tell, this is essentially a transitive inner set-model, but isn’t the existence of such a thing unprovable in ZFC? I haven’t gotten very far through the book yet, but it *seems* to me that there is no loss in reading “small set” as “any set” and non-small sets as proper classes…

- All models of $\mathsf{ZFC}$ between $V$ and $V$ are generic extensions of $V$
- The Continuum Hypothesis & The Axiom of Choice
- Grothendieck Topology on the Category of Elements
- What's the last step in this proof of the uniqueness of equalizers?
- Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?
- The class of all functions between classes (NBG)
- What is the canonical morphism in a category where finite products and coproducts exist?
- Do finite products commute with colimits in the category of spaces?
- How do I get the existence of a set in ZFC following Jech?
- The “set” of equivalence classes of things.

In *Categories for the working mathematician*, Mac Lane assumes the existence of one Grothendieck universe $U$, and $\mathbf{Set}$ is the category of sets in $U$. This device ensures the existence of functor categories like $[\mathbf{Set}, \mathbf{Set}]$.

On the other hand, $\mathbf{Ens}$ is any full subcategory of the metacategory of *all* sets, with the restriction that the collection of objects in $\mathbf{Ens}$ is itself a set. Note that $\mathbf{Ens}$ may fail to have the properties expected of $\mathbf{Set}$, e.g. cocompleteness, cartesian closedness, etc.

- Behavior at $0$ of a function that is absolutely continuous on $$
- Closed form of $\int_0^{\pi/2} \frac{\arctan^2 (\sin^2 \theta)}{\sin^2 \theta}\,d\theta$
- How to find points of tangency on an ellipse?
- What does “curly (curved) less than” sign $\succcurlyeq$ mean?
- Proving sets are measurable
- How to create an identity for $\sin \frac{x}{4}$
- Can different choices of regulator assign different values to the same divergent series?
- Every maximal ideal is principal. Is $R$ principal?
- Simplifying an Arctan equation
- Is the splitting field equal to the quotient $k/(f(x))$ for finite fields?
- proof of $1^4+2^4+…+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$
- Finding the dual cone
- What's the correct naming and notation those algebraic structures?
- sum of squares of dependent gaussian random variables
- Randomly select $k$ boxes and place a ball into the least-loaded one.