A note on topological ℎ𝑜𝑚-functors

This note establishes internal criteria on a category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a separator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which characterize the condition that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-induced covariant hom-functor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript normal upper Sigma Baseline colon upper C right-arrow"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>h</mml:mi> <mml:mi mathvariant="normal">Σ</mml:mi> </mml:msub> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{h_\Sigma }:C \to</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Set is (epi, mono-source)-topological.