The tight groupoid of the inverse semigroups of left cancellative small categories
<p>We fix a path model for the space of filters of the inverse semigroup <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S Subscript normal upper Lamda">
  <mml:semantics>
    <mml:msub>
      <mml:mrow class="MJX-TeXAtom-ORD">
        <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi>
      </mml:mrow>
      <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
    </mml:msub>
    <mml:annotation encoding="application/x-tex">\mathcal {S}_\Lambda</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> associated to a left cancellative small category <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
  <mml:semantics>
    <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi>
    <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>. Then, we compute its tight groupoid, thus giving a representation of its <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk">
  <mml:semantics>
    <mml:msup>
      <mml:mi>C</mml:mi>
      <mml:mo>∗<!-- ∗ --></mml:mo>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">C^*</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-algebra as a (full) groupoid algebra. Using it, we characterize simplicty for these algebras. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.</p>