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Given two locally compact spaces $X$ and $Y$ then the product $X \times Y$ is an open subset inside $\overline{X} \times \overline{Y}$, where $\overline{X}$ and $\overline{X}$ are the one point compac …

First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works. Here are some o …

I've encountered that condition a few time. Here is what I know about it: If that property is satisfied for a model category $M$ then the full subcategory $C$ of finite $I$-cell complex is itself a Br …

You can always rewrite a subobject $V \subseteq \mathbb{Q}$ as a function $\mathbb{Q} \to \Omega$, but you'll need to includes all the axiom that are in the definition. Even if you only look at defini …

The question is basically "do we really have a good way to talk about large categories internally in an elementary topos?" An internal small category in a topos $E$ is just a category object in $E$. U …

I'm stating here what I think is the correct version of the conjecture in John Baez's answer. The 1-categorical theory of rigs has morphisms given by polynomials whose coefficients are in $\mathbb{N}$ …

As far as I'm aware, no such conditions is known - The paper of Anel and Lejay is the closest to an answer available in the litterature. So, this is not an answer to the question, but more of an expan …

Edit : I should clarify that I've interpreted "Etale topos" to mean the petit/small étale topos everywhere. What I've said about Grothendieck-Galois duality only apply to the petit étale topos. If yo …

Here is what I think is the simplest strategy. I'm only giving a sequence of lemma which lead to the result and I think they are all easy enough, but maybe a little teddious to write down (but let me …

If the absence of adjoints is what worries you, you can consider this to be a two-step process - and I would argue that in practice this is the case in the vast majority of cases: One first generalize …

$\DeclareMathOperator\Lex{Lex}\DeclareMathOperator\Delex{Delex}\newcommand\Set{\mathrm{Set}}$If you take $D$ to be the category of sets, you get that $\Lex(C,\Set)$ is the category of functor $[\Delex …

From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is me …

$\DeclareMathOperator\Aut{Aut}\newcommand\card[1]{\lvert#1\rvert}$So, after going over the classification of finite abelian groups in a class I was teaching this winter, I got curious about whether it …

I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably …

Let $C$ be a monoidal category, $X \in C$ an object, and $TX$ the free monoid object on $X$, assuming it exists. How often do we have an isomorphism $TX \otimes X = X \otimes TX$? is there a canonical …