Amalgamable diagram shapes
Abstract
A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category $\mathbf I$, the following are equivalent: (i) every $\mathbf I$shaped diagram in a category with the AP and the JEP has a cocone; (ii) every $\mathbf I$shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category $\mathcal{L}(\mathbf{I})$ of "paths" in $\mathbf I$ has only idempotent endomorphisms. When $\mathbf I$ is a finite poset, these are further equivalent to: (iv) every upwardclosed subset of $\mathbf I$ is simplyconnected; (v) $\mathbf I$ can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite $\mathbf I$.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 DOI:
 10.48550/arXiv.1606.06777
 arXiv:
 arXiv:1606.06777
 Bibcode:
 2016arXiv160606777C
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Logic;
 18A30;
 03C30
 EPrint:
 14 pages