# What is the name of this functor's property?

Assume there is a functor $$L$$ from a category $$C$$ to a category $$D$$ which satisfies the following property: for any objects $$X,Y,Z$$ from $$C$$ and morphisms $$f\colon X\to Y, g\colon X\to Z$$ such that $$L(g)=\varphi\circ L(f)$$ for some morphism $$\varphi\colon L(Y)\to L(Z)$$ there is a morphism $$h\colon Y\to Z$$ such that $$\varphi=L(h)$$ and $$g=h\circ f$$. What is the name of this property?

• Do you require uniqueness of $h$? Commented Mar 19 at 8:38
• Both cases are ok for me
– 8k14
Commented Mar 19 at 8:48
• In model theory the dual of this property ($L(g) = L(f) \circ \varphi$ implies there is $h$ s.t. $L(h) = \varphi$ and $g = f \circ h$) is sometimes called "coherence". So ...... "herence"? Commented Mar 19 at 9:11
• @Mark Kamsma Thanks. Thus "coherence" of $L$ is equivalent to the fact that all the morphisms in $C$ are $L$-cartesian?
– 8k14
Commented Mar 19 at 9:51
• @8k14 It seems so, though I have not seen the uniqueness requirement before (and I must admit that I had never heard of the term "$L$-cartesian"), but it would be automatic in most model-theoretic contexts. Usually the model-theoretic context is where $L$ is an inclusion of categories (not necessarily full, of course). Even more specifically: $D$ could be thought of as the category of structures with embeddings and $C$ the category of models of some theory with elementary embeddings, and $L$ is then the obvious inclusion. Commented Mar 19 at 11:24

In case we require this $$h$$ to be unique, then the property that you wrote down is saying that $$f$$ is an $$L$$-cocartesian morphism (see Definition 2.1 here), so $$L$$ is a functor for which every morphism in $$C$$ is $$L$$-cocartesian. On its own, this does not bring us much (except that all fibers of $$L$$ are groupoids), but we could additionally ask for the following: for every morphism $$a:A\to B$$ in $$D$$ and every object $$A'$$ in $$C$$ such that $$LA'=A$$, there exists a morphism $$a'\colon A'\to B'$$ in $$C$$ such that $$La'=a$$. If this is also satisfied, then your functor $$L$$ is a Grothendieck opfibration fibered in groupoids, also called a left fibration in more modern higher category theory. (By the Grothendieck construction, this would mean that $$L$$ classifies a $$(2,1)$$-functor $$D\to\mathsf{Grpd}$$, where $$\mathsf{Grpd}$$ is the $$(2,1)$$-category of groupoids. Informally, this functor is of the form $$d\mapsto L^{-1}(d)$$.)
If we do not require $$h$$ to be unique, then I am not aware of any definition that talks about this.
• Thanks so much. My functor does not satisfy the extra property you describe but every morphism in $C$ is also $L$-cartesian. Does it give anything?
• Not on its own, as far as I know. Which functor are you exactly considering? It does hold that a Grothendieck opfibration that is also a Grothendieck fibration with the one-arrow category $0\to 1$ as target encodes an adjunction, so maybe you can restrict $D$ to a subcategory on which $L$ is such a functor, and then we would get a whole bunch of adjunctions encoded by $L$. Commented Mar 19 at 17:10