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In the Axiom of Replacement, the "definition" involved is allowed to use parameters, i.e. it is allowed to refer to specific sets. In this case, the functor $F$ is a set, and so your definition using …

Yes, certainly. For instance, you could define $F(t)=a$ for $a<1$ and $F(1)=b$, with the obvious choice of morphisms (the identity whenever possible, and otherwise $m$). More generally, if $I=A\cup …

Not in general, even if by "small" you mean "essentially small" (i.e., there is a small set of isomorphism classes of objects). First of all, it is possible for a single object to have a proper class …

Yes, this is true (up to natural isomorphism). The simplest way to see this is just to write down what the copowers are explicitly. Let us take a skeleton of $\mathcal{V}$ consisting of all vector s …

No such right adjoint exists, even restricted to sober spaces. For simplicity let us take $S=\operatorname{Spec} k$ for some field $k$, and consider the space $X$ having two points, one of which is c …

The condition $\varphi(a)\notin\mathfrak{r}\iff \psi(a)\notin\mathfrak{r}$ is equivalent to $\varphi(a)=\psi(a)$, since an element of a Boolean algebra is determined by the prime ideals that contain i …

As far as I can tell you are correct. In fact, any map in the category of simplices coming out of a nondegenerate simplex must be injective. If $a:\Delta^n\to X$ is a nondegenerate simplex and $f:\D …

Certainly not in general. For instance, let $F$ be the inclusion of the category of finite-dimensional vector spaces (over some fixed field) into the category of all vector spaces. If you want $\ma …

It is trivial that such a maximal $E$ exists, and consists of pairs $\{f,g\}$ such that whenever $x$ and $y$ are adjacent, $f(x)$ and $g(y)$ are adjacent. However, it does not make $\operatorname{Hom …

When you pass to a larger universe, all categories that were in your old universe become small. A small category which is not a poset cannot be closed under colimits (think about taking coproducts wi …

I believe the following is a counterexample. Let $\mathcal{A}$ and $\mathcal{B}$ be closed symmetric monoidal abelian categories such that the unit object $1\in\mathcal{B}$ is projective and let $F:\ …

This seems entirely straightforward unless I'm missing something. For any $n\in\mathbb{N}$, $\Sigma_n$ acts on $X^{\otimes m}$ for any $m\geq n$ (on the first $n$ coordinates), and this action commut …

Suppose you have such a preorder $\leq$. Suppose you have $f,f':x\to y$ such that $f\leq f'$ but $\operatorname{im}(f)\not\subseteq\operatorname{im}(f')$. We can then compose with the quotient $q:y\ …

Here's a nice example that recently came up in an MSE question. Let $k$ be a field and let $Vect$ be the category of $k$-vector spaces and $Aff$ be the category of $k$-affine spaces. Every vector sp …

The assumption that the kernel exists is built into the definition of "termwise split injection" in these notes: Definition 9.4 says A termwise split injection $\alpha:A^\bullet\to B^\bullet$ is a …