The proof that the walking splitting has sifted colimits does not really need the siftedness of the index category; it merely uses that finitely objects have a cocone (in the index category). The same has happened for the walking coreflexive pair (see MO).
This suggests adding this property to the database as well as having colimits of this shape. As usual, everything can be dualized.
In the preprint Amalgamable diagram shapes the property is called the Joint Embedding Property (JEP). I don't like this since "embedding" only makes sense in the context of model theory (where this comes from) where one considers embeddings only. But anyway, let's call the new property "satisfies JEP" if nothing else comes up. Not sure about the dual. If colimits of shape JEP exist, we might say the category "has JEP-colimits".
Thus:
- sifted ===> JEP
- has JEP-colimits ===> has sifted colimits ( ===> has filtered colimits)
- has JEP-colimits ===> has coequalizers
- If $K$ is a field and $d \geq 0$, the category of $K$-vector spaces of dimension $\leq d$ has JEP-colimits.
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$I$ satisfies JEP if and only if $I$ is non-empty and every pair of objects in $I$ has a cospan. In other words, $CoSpan(I)$ is strongly connected.
The property JEP is very interesting when applied to embeddings. A thin category satisfying JEP is precisely a directed preorder.
This issue is resolved when the properties (JEP, JEP-colimits, and the duals) have been added and decided for all categories in the database (except for a few hard cases).
Maybe we can call it the "Discrete Cocone Property" (DCP), or "weakly filtered".
The property has also been mentioned in a comment on the implication "thin_sifted_is_filtered" that states: thin + sifted => filtered. Namely, sifted can be weakened to JEP.
The proof that the walking splitting has sifted colimits does not really need the siftedness of the index category; it merely uses that finitely objects have a cocone (in the index category). The same has happened for the walking coreflexive pair (see MO).
This suggests adding this property to the database as well as having colimits of this shape. As usual, everything can be dualized.
In the preprint Amalgamable diagram shapes the property is called the Joint Embedding Property (JEP). I don't like this since "embedding" only makes sense in the context of model theory (where this comes from) where one considers embeddings only. But anyway, let's call the new property "satisfies JEP" if nothing else comes up. Not sure about the dual. If colimits of shape JEP exist, we might say the category "has JEP-colimits".
Thus:
The property JEP is very interesting when applied to embeddings. A thin category satisfying JEP is precisely a directed preorder.
This issue is resolved when the properties (JEP, JEP-colimits, and the duals) have been added and decided for all categories in the database (except for a few hard cases).
Maybe we can call it the "Discrete Cocone Property" (DCP), or "weakly filtered".
The property has also been mentioned in a comment on the implication "thin_sifted_is_filtered" that states: thin + sifted => filtered. Namely, sifted can be weakened to JEP.