Twisted double functors and
loosely discrete opfibrations
Category Theory 2026
2026-07-17
The tight Hom functor
The (tight) Hom functor on a double category \(\mathbb{D}\) is a lax double functor
\[ \operatorname{Hom}_{\mathbb{D}}: \mathbb{D}^\mathrm{op}\times \mathbb{D} \to \mathbb{S}\mathsf{et} \]
that sends
- a pair of objects \((x,y)\) to the set \(\mathbb{D}_0(x,y)\) of tight morphisms \(x \to y\)
- a pair of tight morphisms \((f: x' \to x,\ g: y \to y')\) to the function \[ g \circ (-) \circ f: \mathbb{D}_0(x,y) \to \mathbb{D}_0(x',y') \]
a pair of loose morphisms \((m, n)\) to the span with apex \(\mathbb{D}_1(m,n)\):
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The twisted Hom functor (2)
a pair of tight morphisms \((f: x \to w,\ g: y \to z)\) to the profunctor
\[ \mathbb{D}(f,g): \mathbb{D}(x,y) \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}\mathbb{D}(w,z) \]
with heteromorphisms
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and actions by pre-/post-composing with globular cells
Twisted lax functors
Definition (sketch). A twisted lax functor \(F: \mathbb{D} \looparrowright \mathbb{E}\) sends
- an object \(x\) of \(\mathbb{D}\) to an object \(Fx\) of \(\mathbb{E}\)
- a tight morphism \(f: x \to y\) to a loose morphism \(Ff: Fx \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}Fy\)
- a loose morphism \(m: x \mathrel{\mkern 3mu\vcenter{\hbox{$\scriptstyle+$}}\mkern-13mu{\to}}y\) to a tight morphism \(Fm: Fx \to Fy\)
a cell \(\alpha\) of \(\mathbb{D}\) to a cell \(F\alpha\) of \(\mathbb{E}\) of transposed shape:
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and contains the following comparison cells…
Twisted lax functors (2)
tight-to-loose composition and identity comparisons
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loose-to-tight composition and identity comparisons
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assumed to be invertible (w.r.t. loose composition).
Recall: internal discrete opfibrations
Let \(\mathsf{C}\) be a category with pullbacks.
Definition. A functor \(P: \mathsf{E} \to \mathsf{B}\) internal to \(\mathsf{C}\) is a discrete opfibration if the canonical map \(\langle d_0, P_1 \rangle\) is an isomorphism:
Problem: Naively instantiating in \(\mathsf{C} = \mathsf{Cat}\) asks that \(\langle d_0, P_1 \rangle\) be an isomorphism of categories—too strict!
Loosely discrete opfibrations
Definition. A double functor \(P: \mathbb{E} \to \mathbb{B}\) is a loosely discrete opfibration (ldopf) if the canonical functor \(\langle \operatorname{src}, P_1 \rangle\) is an equivalence of categories:
- fully faithful \(\implies\) uniqueness of opcartesian lifts (up to iso)
- essentially surjective \(\implies\) existence of lifts (up to iso)
Fact. \(P\) is locally discrete in the loose direction: each fiber \(\mathbb{E}_x \mathrel{\vcenter{:}}=P^{-1}(x)\), \(x \in \mathbb{B}\), is (equivalent to) a loosely discrete double category.
Cloven loosely discrete opfibrations
Definition. A cleavage for an ldopf \(P: \mathbb{E} \to \mathbb{B}\) is a functor
\[ \sigma: \mathbb{E}_0 \times_{\mathbb{B}_0} \mathbb{B}_1 \to \mathbb{E}_1 \]
that is left-adjoint-right-inverse (LARI) to the canonical functor \(\langle \operatorname{src}, P_1 \rangle\)
along with
a pseudo-algebra of the pull-push pseudo-monad given by \(\mathbb{B}\), having carrier and structure map
\[(P_0: \mathbb{E}_0 \to \mathbb{B}_0,\ \operatorname{tgt}\sigma: \mathbb{E}_0 \times_{\mathbb{B}_0} \mathbb{B}_1 \to \mathbb{E}_0)\]
coherently with the counit \(\varepsilon\) of the adjunction \(\sigma \dashv \langle \operatorname{src}, P_1 \rangle\).
Elements correspondence
Theorem. For any double category \(\mathbb{B}\), there is a correspondence
in which the forward direction is an elements construction.
To be precise, these constructions round-trip, up to:
- for each twisted copresheaf \(T: \mathbb{B} \looparrowright\mathbb{C}\mathsf{at}\), an invertible natural transformation \[ \gamma: T \Rightarrow\mathsf{Tw}(\mathbb{E}\mathsf{lt}(T)) \]
for each cloven ldopf \(P: \mathbb{E} \to \mathbb{B}\), an equivalence of double categories over \(\mathbb{B}\)
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