Barr and Wells (Toposes, Triples, and Theories, 84) talks about arrows as a general kind of elements. In $\mathbf{Set}$, arrows from $\{\ast\}\to A$ are the usual elements of $A$, and arrows from ...

To the best of my knowledge, no abstract formulation and soundness theorem of the full Penrose notation (with symmetriser, antisymmetriser, covariant derivative, etc) exists. There is, however, a ...

In general, there are probably no conditions for the existence of natural transformations that are simpler than just using the definition of naturality itself. In a category Z with a zero object, the ...

In categorical literature, the notation $f: X \rightarrow Y$ only means "$f$ is a function from $X$ to $Y$" in the case where the category you are working in is Set, that is the category of sets and ...

For figures in tex papers, pgf/tikz is usually my go-to package. However, if interactively is a concern, this is certainly not the way to go (tweak, build, tweak, build, ....). It can certainly do ...

This is true in boolean categories (extensive + terminal object + (T : 1 → 1 + 1) is subobject classifier), but for any monic, not just split monics. There's quite a nice and easy read on ...

I've been working on a GUI for typesetting tensor/monoidal diagrams in TikZ. http://tikzit.sourceforge.net/ It's especially geared at applications to quantum mechanics, namely "dot"-style diagrams ...

Relations are a toy model for linear maps. In fact, they can be thought of as matrices over the boolean semi-ring.

Paul-André Melliès has quite an interesting paper on this topic: http://hal.archives-ouvertes.fr/docs/00/33/93/31/PDF/free-models.pdf ...but phrased in the more general terms of T-algebras of a ...

One way to think of categories algebraically is as a "monoid-oid". So, do for monoids what you do for groups to make groupoids. A (small) category is a set, with an associative, unital, partial ...