In this paper‎, ‎we give the notions of crossed polymodule and cat$^1$-polygroup‎ ‎as a generalization of Loday's definition‎. ‎Then‎, ‎we define the pullback cat$^1$-polygroup and we obtain some results in this respect‎. ‎Specially‎, ‎we prove that by a pullback cat$^1$-polygroup we can obtain a cat$^1$-group‎.

A 7 – year - old Caucasian girl presented with a 3-month history of nodules on her thighs and buttocks. During a one-month period, the lesions gradually increased in size and number. They became indurate and tender. Similar nodules appeared on forearms after 4 months. The lesions gradually healed with a depressed scar. On physical examination, we found many indurate areas 10-40 mm in diam...

A 10-year-old girl presented with a 1-year history of an eruption on her hands. Physical examination revealed soft, bluish slightly tender subcutaneous nodules and swellings in the ulnar site and palmar aspect of five fingers of the right hand and dorsal aspect of three fingers of the left hand without any gross deformity in fingers and long bones (figure 1). History of pathological fract...

A 55-year-old woman was referred to our outpatient clinic because of bilateral papular lesions on her nose from 2 years ego. In clinical examination, we found three symmetrical dome-shaped, skin-colored, asymptomatic papules on both nasal alas. The lesions made the patient's nose to seem larger with eroded edges (figures 1, 2). Examination of nasal mucosa was normal. General physical examinatio...