The category **RefGph** of [reflexive directed graphs](https://ncatlab.org/nlab/show/reflexive+graph) is the functor
category $\hat{∆}\_1=\mbox{Fun}(∆^◦\_1,$**Set**), where $∆\_1$ is 
the [simplex category](https://ncatlab.org/nlab/show/simplex+category) truncated at level 1. 


[Hence](https://ncatlab.org/nlab/show/Heyting+category) the poset Sub(X) of subobjects of such a graph X is a [Heyting algebra](https://ncatlab.org/nlab/show/Heyting+algebra).


I am interested in the category $K\_1$, whose objects are the Heyting algebras, which are isomorphic to those of the form Sub(X) for some reflexive directed graph X and in the corresponding superintuitionistic propositional logic for which these Heyting algebras are algebraic models i.e. in the set $L(K\_1)$, constisting of those [propositional formulas](https://en.wikipedia.org/wiki/Propositional_formula) $\varphi$ in the variables $p\_i$, which are valid in exactly the objects of $K\_1$. (graph logic)


In particular I would like answers or explanations to the following questions: 


1. How does one construct (finite) [coproducts](https://ncatlab.org/nlab/show/coproduct) in $K\_1$ explicitly?
2. How does one describe [free objects](https://ncatlab.org/nlab/show/free+object) of $K\_1$ on finite generating sets explicitly?
3. Can the objects of the category $K\_1$ be characterized by a finite set of explicit identities?
4. Can the propositional logic $L(K\_1)$ be described by a finite set of explicit axioms?


Can you give some references to relevant papers?


May be you even have ideas concerning the following question:


5. Can you think of an [interpretation](https://en.wikipedia.org/wiki/Brouwer%E2%80%93Heyting%E2%80%93Kolmogorov_interpretation) of the logic $L(K\_1)$ as a special type of constructive reasoning?



Heyting algebras originating from directed graphs
These can be described in a manner similar to my answer to [this related MO question](https://mathoverflow.net/questions/159989/internal-logic-of-the-topos-of-simplicial-sets) about simplicial sets. This case is simpler and to understand the subobject classifier in the topos of reflexive graphs, it suffices to look at the lattice of abstract simplicial complexes on $\{0,1\}$ (including the empty one):


* $1 = \{\{0,1\},\{0\},\{1\}\}$
* $s \vee t = \{\{0\},\{1\}\}$
* $s = \{\{0\}\}$
* $t = \{\{1\}\}$
* $0 = \varnothing$


Let's write $\Omega\_1$ for this Heyting algebra.


In down to earth terms, this lattice represents all five possibilities of what could happen to a non-loop edge $e$ of $X$ in a reflexive subgraph $X'$:


* $e$ is an edge in $X'$
* $e$ is not an edge in $X'$ but both its source and target vertices remain in $X'$
* $e$ is not an edge in $X'$ and only its source vertex remains in $X'$
* $e$ is not an edge in $X'$ and only its target vertex remains in $X'$
* $e$ is not an edge in $X'$ and its source and target vertices are not in $X'$


We now see that $\operatorname{Sub}(X)$ is naturally a subalgebra of a power of $\Omega\_1$, specifically the subalgebra of all $p \in \operatorname{Edges}(X)\to\Omega\_1$ which are coherent (e.g. if $e$ is a loop then $p(e) = 0$ or $p(e) = 1$ and if two edges $e,e'$ share a vertex then $p(e)$ and $p(e')$ shouldn't disagree on the fate of that shared vertex).


The logic $L(K\_1)$ is precisely that of $\Omega\_1$. I'm not sure what to make of your other questions about $K\_1$. The main issue is that the Heyting algebras $\operatorname{Sub}(X)$ are always complete, so it's not clear that free objects should exist in $K\_1$ unless I'm missing something.


