Posts with this logo use 
January 29, 2025
Comagnitude 2
Posted by Tom Leinster
Previously: Part 1
Last time, I talked about the magnitude of a set-valued functor. Today, I’ll introduce the comagnitude of a set-valued functor.
I don’t know how much there is to the comagnitude idea. Let’s see! I’ll tell you all the interesting things I know about it.
Along the way, I’ll also ask an elementary question about group actions that I hope someone knows how to answer.
January 22, 2025
Comagnitude 1
Posted by Tom Leinster
Next: Part 2
In this post and the next, I want to try out a new idea and see where it leads. It goes back to where magnitude began, which was the desire to unify elementary counting formulas like the inclusion-exclusion principle and the simple formula for the number of orbits in a free action of a group on a finite set.
To prepare the ground for comagnitude, I need to present magnitude itself in a slightly different way from usual. I won’t assume you know anything about magnitude, but if you do, watch out for something new: a connection between magnitude and entropy (ordinary, relative and conditional) that I don’t think has quite been articulated before.
January 15, 2025
The Dual Concept of Injection
Posted by Tom Leinster
We’re brought up to say that the dual concept of injection is surjection, and of course there’s a perfectly good reason for this. The monics in the category of sets are the injections, the epics are the surjections, and monics and epics are dual concepts in the usual categorical sense.
But there’s another way of looking at things, which gives a different answer to the question “what is the dual concept of injection?”