The Unapologetic Mathematician

Mathematics for the interested outsider

Diagram categories

A light post today, as I finish up my packing and get ready to head out tomorrow.

Diagram categories are one of those things that at first blush seem almost trivial, but they turn out to be very useful. In general, we start with some small category \mathcal{D} that describes the form of a diagram, and then we take the category of functors \mathcal{C}^\mathcal{D} into the category we’re interested in studying.

An easy example is a set — a category with nothing but identity arrows. A functor from a set to \mathcal{C} just picks out one object of \mathcal{C} for each element of the set. A little more interesting is the category \bullet\rightrightarrows\bullet. This has two objects and two (nontrivial) morphisms. A functor from this category picks out two objects from \mathcal{C} and two (in general different) parallel arrows from one to the other in \mathcal{C}.

Even more useful is the category
\begin{matrix}\bullet&\rightarrow&\bullet\\\downarrow&&\downarrow\\\bullet&\rightarrow&\bullet\end{matrix}
Of course we can compose the arrows around the upper right or those around the lower left to get diagonal arrows, but we insist here that those two diagonal arrows are the same. A functor from this category picks out four objects and four morphisms that describe a commuting square in \mathcal{C}.

As a bit of a teaser, notice that the setup for finding a binary equalizer in \mathcal{C} is exactly a functor from the category \bullet\rightrightarrows\bullet to \mathcal{C}. Similarly, the setup for a binary product is a functor from the category \bullet\qquad\bullet. What categories give rise to diagram categories for the setups for multiple (co)products? (multiple) (co)equalizers? pushouts? pullbacks? How are the setups for pushouts and pullbacks different from the setups for equalizers and coequalizers, and why?

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June 16, 2007 - Posted by | Category theory

1 Comment »

  1. [...] Cones and cocones There are a few auxiliary concepts we’ll need before the next major topic. Let’s start with two categories and , and the category of functors . If the following seems very complicated, consider to be any particular toy category you’d like, so this category of functors is a diagram category. [...]

    Pingback by Cones and cocones « The Unapologetic Mathematician | June 18, 2007 | Reply


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