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diff --git a/matching/include/simplex.h b/matching/include/simplex.h
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+++ b/matching/include/simplex.h
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+#ifndef MATCHING_DISTANCE_SIMPLEX_H
+#define MATCHING_DISTANCE_SIMPLEX_H
+
+#include <algorithm>
+#include <vector>
+#include <ostream>
+
+#include "common_util.h"
+
+namespace md {
+
+    template<class Real>
+    class Bifiltration;
+
+    enum class BifiltrationFormat {
+        phat_like, rivet
+    };
+
+    class AbstractSimplex {
+    private:
+        std::vector<int> vertices_;
+    public:
+
+        // this member is for convenience only;
+        // abstract simplices are identified  by their set of vertices
+        mutable int id {-1};
+
+        decltype(auto) begin() { return vertices_.begin(); }
+
+        decltype(auto) end() { return vertices_.end(); }
+
+        decltype(auto) begin() const { return vertices_.begin(); }
+
+        decltype(auto) end() const { return vertices_.end(); }
+
+        decltype(auto) cbegin() const { return vertices_.cbegin(); }
+
+        decltype(auto) cend() const { return vertices_.cend(); }
+
+        int dim() const { return vertices_.size() - 1; }
+
+        void push_back(int v)
+        {
+            vertices_.push_back(v);
+            std::sort(vertices_.begin(), vertices_.end());
+        }
+
+        AbstractSimplex() { }
+
+        AbstractSimplex(std::vector<int> vertices, bool sort = true)
+                :vertices_(vertices)
+        {
+            if (sort)
+                std::sort(vertices_.begin(), vertices_.end());
+        }
+
+
+        template<class Iter>
+        AbstractSimplex(Iter beg_iter, Iter end_iter, bool sort = true)
+                :
+                vertices_(beg_iter, end_iter)
+        {
+            if (sort)
+                std::sort(vertices_.begin(), end());
+        }
+
+        std::vector<AbstractSimplex> facets() const
+        {
+            std::vector<AbstractSimplex> result;
+            for (int i = 0; i < static_cast<int>(vertices_.size()); ++i) {
+                std::vector<int> facet_vertices;
+                facet_vertices.reserve(dim());
+                for (int j = 0; j < static_cast<int>(vertices_.size()); ++j) {
+                    if (j != i)
+                        facet_vertices.push_back(vertices_[j]);
+                }
+                if (!facet_vertices.empty()) {
+                    result.emplace_back(facet_vertices, false);
+                }
+            }
+            return result;
+        }
+
+        friend std::ostream& operator<<(std::ostream& os, const AbstractSimplex& s);
+        // compare by vertices_ only
+        friend bool operator==(const AbstractSimplex& s1, const AbstractSimplex& s2);
+
+        friend bool operator<(const AbstractSimplex&, const AbstractSimplex&);
+    };
+
+    inline std::ostream& operator<<(std::ostream& os, const AbstractSimplex& s)
+    {
+        os << "AbstractSimplex(id = " << s.id << ", vertices_ = " << container_to_string(s.vertices_) << ")";
+        return os;
+    }
+
+    inline bool operator<(const AbstractSimplex& a, const AbstractSimplex& b)
+    {
+        return a.vertices_ < b.vertices_;
+    }
+
+    inline bool operator==(const AbstractSimplex& s1, const AbstractSimplex& s2)
+    {
+        return s1.vertices_ == s2.vertices_;
+    }
+
+    template<class Real>
+    class Simplex {
+    private:
+        Index id_;
+        Point<Real> pos_;
+        int dim_;
+        // in our format we use facet indices,
+        // this is the fastest representation for homology
+        // Rivet format fills vertices_ vector
+        // Simplex alone cannot convert from one representation to the other,
+        // conversion routines are in Bifiltration
+        Column facet_indices_;
+        Column vertices_;
+        Real v {0}; // used when constructed a filtration for a slice
+    public:
+        Simplex(Index _id, std::string s, BifiltrationFormat input_format);
+
+        Simplex(Index _id, Point<Real> birth, int _dim, const Column& _bdry);
+
+        void init_rivet(std::string s);
+
+        void init_phat_like(std::string s);
+
+        Index id() const { return id_; }
+
+        int dim() const { return dim_; }
+
+        Column boundary() const { return facet_indices_; }
+
+        Real value() const { return v; }
+
+        // assumes 1-criticality
+        Point<Real> position() const { return pos_; }
+
+        void set_position(const Point<Real>& new_pos) { pos_ = new_pos; }
+
+        void scale(Real lambda)
+        {
+            pos_.x *= lambda;
+            pos_.y *= lambda;
+        }
+
+        void translate(Real a);
+
+        void set_value(Real new_val) { v = new_val; }
+
+        friend Bifiltration<Real>;
+    };
+
+    template<class Real>
+    std::ostream& operator<<(std::ostream& os, const Simplex<Real>& s);
+
+}
+
+#include "simplex.hpp"
+
+#endif //MATCHING_DISTANCE_SIMPLEX_H