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 ...

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 ...

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.

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 ...

To follow on from what Qiaochu said, one of the interesting things about groupoids is their cardinality. Whereas the cardinality of a set is a natural number, the cardinality of a groupoid is 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 ...

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 ...

I've found a major stumbling block when trying to introduce category theory to new people is a tendency to think of objects as "things" instead of "types of things". So, after coping just fine with ...

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 ...