Search Results

In general $F$ preserves neither pullbacks nor even products. In a comment I mentioned that the "discrete graph" functor $\text{Set} \to \text{Set}^{\bullet \rightrightarrows \bullet}$ is full and fai …

Quoting the nLab article on accessible categories: $C$ is accessible iff one of the following equivalent conditions holds: it is the category of models (in $\mathbf{Set}$) of some small sketch. i …

Edit: The following is a second attempt to repair problems in earlier proposed solutions, as pointed out by Gerald Edgar in comments. Hopefully this time I've gotten it right this time. It's classic …

Also read some of the original papers by Baez and Dolan, for example Categorification (in Higher Category Theory, Contemporary Mathematics 230, AMS 1998), where the Tangle Hypothesis is explained amon …

"Relatively free" functors have been considered at least since the time of Lawvere's thesis; see for example page 111 of 122 for the case involving finitary algebraic theories. I don't know where th …

By $\widehat{\mathcal{A}}$ you must mean the category of presheaves on $\mathcal{A}$. I take it that $X$ denotes a presheaf, and that $\mathcal{A}/X$ is the comma category $y_{\mathcal{A}} \downarrow …

As Andreas Blass points out, it depends partly on how you are defining affine spectra, but I presume you are considering them as locally ringed spaces of suitable sort. Have you consulted the followin …

I will jot down some stray (and easy) thoughts, in an effort to engage the question and see whether some aspects of it can be made more precise. Going out on a limb, I suppose a baseline assumption …

Conway had an analysis of the notorious Steiner-Lehmus theorem, arguing that no "equality-chasing proof" is possible. MO user Timothy Chow initiated a discussion about Conway's analysis on the FOM lis …

Four ants are on the corners of a square, each facing its neighbor in the counterclockwise direction. At the same time each begins marching towards its neighbor, all at the same speed. After moving th …

(This answer is Community Wiki and is extracted from comments by user eric, who has declined to post them as an answer. The CW is to invite others to contribute, especially to provide suitable referen …

In 2-category theory, there is a notion of "lax idempotent 2-monad" $M$ on a 2-category, for which the multiplication $m: M M \to M$ is left adjoint to the unit $u M: M \to M M$. A typical sort of exa …

I'm also not quite sure what the question is, but I'll answer anyway with the hope of saying something useful. The notion of closed idempotent referred to above is the same as the notion of idempote …

What you are describing is an example of Max Kelly's notion of club, closely connected with the concept of operad. The original references date back to the 70's; one reference is G.M.Kelly. On club …

If you are disallowing multiple edges between vertices, then such graphs are the same things as binary relations $R$ on the vertex set (where $x R y$ iff there is an edge from $x$ to $y$. Then $M$ wou …