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W1977823902 abstract "We give a simple characterization of full subcategories of equational categories. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {a}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is one such and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper B> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>B</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the category of topological spaces, we consider a pair of adjoint functors <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a Superscript o p Baseline long left-right-arrow Overscript upper U Underscript upper F Endscripts script upper B> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>o</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:munder> <mml:mover> <mml:mo stretchy=false>⟷</mml:mo> <mml:mi>U</mml:mi> </mml:mover> <mml:mi>F</mml:mi> </mml:munder> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>B</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {a}^{op}}underset {F}{overset {U}{longleftrightarrow }}mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are represented by objects <italic>I</italic> and <italic>J</italic> in the sense that the underlying sets of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>U(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F left-parenthesis upper B right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F(B)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a left-parenthesis upper A comma upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {a}(A,I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper B left-parenthesis upper B comma upper J right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>B</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {B}(B,J)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (One may take <italic>I</italic> and <italic>J</italic> to have the same underlying set.) Such functors always establish a duality between Fix <italic>FU</italic> and Fix <italic>UF</italic>. We study conditions under which one can conclude that <italic>FU</italic> and <italic>UF</italic> are reflectors into Fix <italic>FU</italic> and Fix <italic>UF</italic>, that Fix <italic>FU</italic> = Image <italic>F</italic> = the limit closure of <italic>I</italic> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {a}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that Fix <italic>UF</italic> = Image <italic>U</italic> = the limit closure of <italic>J</italic> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper B> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>B</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For example, this happens if (1) <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {a}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a limit closed subcategory of an equational category, (2) <italic>J</italic> is compact Hausdorff and has a basis of open sets of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartSet x element-of upper J vertical-bar alpha left-parenthesis upper I right-parenthesis left-parenthesis x right-parenthesis not-equals beta left-parenthesis upper I right-parenthesis left-parenthesis x right-parenthesis EndSet> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>J</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mi>β</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{ x in J|alpha (I)(x) ne beta (I)(x)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=application/x-tex>alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta> <mml:semantics> <mml:mi>β</mml:mi> <mml:annotation encoding=application/x-tex>beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are unary <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {a}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operations, and (3) there are quaternary operations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ</mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=eta> <mml:semantics> <mml:mi>η</mml:mi> <mml:annotation encoding=application/x-tex>eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that, for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x element-of upper J Superscript 4 Baseline comma xi left-parenthesis upper I right-parenthesis left-parenthesis x right-parenthesis equals eta left-parenthesis upper I right-parenthesis left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>J</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>η</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>x in {J^4},xi (I)(x) = eta (I)(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 1 equals x 2> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{x_1} = {x_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 3 equals x 4> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{x_3} = {x_4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (The compactness of <italic>J</italic> may be dropped, but then one loses the conclusion that Fix <italic>FU</italic> is the limit closure of <italic>I</italic>.) We also obtain a quite different set of conditions, a crucial one being that <italic>J</italic> is compact and that every <italic>f</italic> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper B left-parenthesis upper J Superscript n Baseline comma upper J right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>B</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>J</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {B}({J^n},J)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>n</italic> finite, can be uniformly approximated arbitrarily closely by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script a> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>a</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {a}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operations on <italic>I</italic>. This generalizes the notion of functional completeness in universal algebra. The well-known dualities of Stone and Gelfand are special cases of both situations and the generalization of Stone duality by Hu is also subsumed." @default.
W1977823902 created "2016-06-24" @default.
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W1977823902 date "1979-01-01" @default.
W1977823902 modified "2023-09-25" @default.
W1977823902 title "A general Stone-Gel’fand duality" @default.
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