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Stream: theory: category theory

Topic: monad not preserving epimorphisms

Tom Hirschowitz (May 20 2022 at 10:30):

What's an example of a monad not preserving epimorphisms? I'm also interested in the same question with other kinds of epimorphisms (strong, regular,...).

Tom Hirschowitz (May 20 2022 at 10:50):

Oh, sorry, that's easy: the free category monad on graphs. $[1]+[1] \to [2]$ is epi, but its image is not. (Although it is as a functor...)

Morgan Rogers (he/him) (May 20 2022 at 10:54):

Consider the monad on pointed sets defined on objects by $A \mapsto A \vee A$ (wedge product, obtained from disjoint union by identifying basepoints; this is the coproduct of pointed sets), with $f:A \to B$ mapped to $f \vee 1$ ($f$ on the left-hand component, everything in the right-hand component mapped to the base-point). We have unit $\iota_1: A \to A \vee A$ and $(\iota_1 \vee \iota_2) \vee (\iota_2 \vee \iota_2) : (A \vee A) \vee (A \vee A) \to A \vee A$ the multiplication.

Morgan Rogers (he/him) (May 20 2022 at 11:18):

Tom Hirschowitz said:

Oh, sorry, that's easy: the free category monad on graphs. $[1]+[1] \to [2]$ is epi, but its image is not. (Although it is as a functor...)

What constraint are you putting on graphs here? If you mean over the category of directed graphs, the inclusion of the vertices into the graph with one edge is not epic

Morgan Rogers (he/him) (May 20 2022 at 11:19):

(because I have two homomorphisms from $\bullet \rightarrow \bullet$ to $\bullet \rightrightarrows \bullet$)

Reid Barton (May 20 2022 at 11:19):

I think $[n]$ is a path of $n$ edges, topologist-style.

Reid Barton (May 20 2022 at 11:20):

Another example: We can take the reader monad $TX = X^A$. This doesn't work in Set because Set satisfies AC, but it could work in another topos. For instance, we could take $TX = X^{\Delta^1}$ on simplicial sets.

Morgan Rogers (he/him) (May 20 2022 at 11:20):

Ah I see, sure. Nice example too, @Reid Barton

Reid Barton (May 20 2022 at 11:21):

We need a map $q : X \to Y$ of simplicial sets which is an epimorphism but for which $q^{\Delta^1}$ is not epi. For example, we can take $Y = \Delta^1 \times \Delta^1$ and $X$ its cover by two disjoint copies of $\Delta^2$. Then $Y^{\Delta^1}$ has a "characteristic" 1-simplex which does not lift to $X^{\Delta^1}$.