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Madeleine Birchfield
Madeleine Birchfield
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In category theory, a dagger category is a categoryprecategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

and which satisfies an equivalent Segal condition using unitary isomorphisms as categories do using isomorphisms.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

In category theory, a dagger category is a category $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

In category theory, a dagger category is a precategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$

and which satisfies an equivalent Segal condition using unitary isomorphisms as categories do using isomorphisms.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

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Madeleine Birchfield
Madeleine Birchfield
  • 2.6k
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  • 7
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In category theory, a dagger category is a precategorycategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

In category theory, a dagger category is a precategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

In category theory, a dagger category is a category $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

category -> precategory, as the simplicial set of dagger categories do not satisfy the Segal condition which is satisfied in categories.
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Madeleine Birchfield
Madeleine Birchfield
  • 2.6k
  • 1
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In category theory, a dagger category is a categoryprecategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

In category theory, a dagger category is a category $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

In category theory, a dagger category is a precategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

  • for all objects $A:\mathcal{C}$, $\mathrm{id}_A^{\dagger_{A,A}} = \mathrm{id}_A$,
  • for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^{\dagger_{A,C}} = f^{\dagger_{A,B}} \circ g^{\dagger_{B,C}}$
  • ${(f^{\dagger_{A,B}})}^{\dagger_{B,A}} = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?

added 56 characters in body
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Madeleine Birchfield
Madeleine Birchfield
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Source Link
Madeleine Birchfield
Madeleine Birchfield
  • 2.6k
  • 1
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  • 23