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Yesterday I observed (and proved) the following odd fact, which I found very surprising. I'm very curious to know if this was known by some people, or if it follows from some other more general fact, …

Small and nicely small are indeed equivalent. I would consider this a fairly classical observation in the topic of small functor, at least when $M = Set$, so I wouldn't be surprised that Day and Lack …

They are completely different constructions which are in general absolutely unrelated but might coincide on some rare occasion. the image of closed immersion vs open immersion is quite good I think, e …

A first remark is that question 1 and 2 are equivalent as in the category pra you have $\Delta_f \dashv \Pi_f$. So if you have a characterization of one class you characterize the other as their left/ …

In short: There is absolutely no such relations, and in general essentially no properties of arrows of the codomain (except being a split epi/split mono or an iso) have any effect on the base change f …

The two definitions are not equivalent: The following counter-example might not be the simplest, and I mostly learned it from Christian Espindola. It is a nice counterexample to quite a lot of similar …

Discrete fibrations are faithful functors, so any discrete fibration to a locally small category is locally small. In particular if you take any comprehension factorization: $$ C \rightarrow D \right …

One way to write the universal property of this object $E(a)$ is as follows: a map $x \to E(a)$ is the same as an isomorphism class of epimorphism $x \times a \twoheadrightarrow k$ in $Set/x$, that i …

Here are three possibilities: First: you have the obvious 'universal solution': You start from $C$ any category and $J$ a set of maps, you can consider the category $C'$ freely generated from $C$ by …

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is isomorphi …

If $A$ and $B$ are small and $D$ is locally presentable then $u$ is a monadic right adjoint. More generally, what is non-trivial is the construction of a left adjoint of $u$. As soon as $u$ has a left …

The following is a completement to the answer by Yonatan. I managed to work it out after I read what he had done. It is basically about proving the converse to the implication he showed in his answer …

The question is extremely dependent on how size issues are handled and many choice that can be made, so that it is very hard to give a general answer. If you really work with general presheaves catego …

I have a complete and co-complete monoidal category in which the tensor product commutes to filtered colimit in each variables, and I would like to understand the "free monoid" construction in this ca …

There is probably no definite answer to this question. But let me propose a few ideas: 1) this kind of general "thesis" only makes sense for naturality with respect to isomorphisms: There is plenty …