nLab cospan

Redirected from "(2,0)-horn category".

Context

Category theory

2-Category theory

(,1)(\infty,1)-Category theory

Internal (,1)(\infty,1)-Categories

Directed homotopy type theory

Contents

Definition

In category theory

In any category, a cospan is a diagram like this:

a b f g c \array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }

A cospan in the category CC is the same as a span in the opposite category C opC^{op}. So, all general facts about cospans in CC are general facts about spans in C opC^{op}, and the reader may turn to the entry on spans to learn more.

A cospan that admits a cone is called a quadrable cospan.

In synthetic (,1)(\infty,1)-category theory

In simplicial type theory, binary parametric observational type theory, and other type theoretic approaches to synthetic (,1)(\infty,1)-category theory, a cospan in a synthetic (,1)(\infty,1)-category CC can be defined as the following:

  • a record consisting of objects (elements of types) x:Cx:C, y:Cy:C, and z:Cz:C and morphisms (elements of hom types) g:hom C(y,z)g:\mathrm{hom}_C(y, z) and h:hom C(x,z)h:\mathrm{hom}_C(x, z)

  • a function k:Λ 0 2Ck:\Lambda_0^2 \to C from the (2,0)-horn type Λ 0 2\Lambda_0^2 to CC, where Λ 0 2\Lambda_0^2 is defined in terms of the directed interval as the type of pairs of elements i,j:𝕀i, j:\mathbb{I} such that i=1i = 1 or j=1j = 1.

Λ 0 2 i:𝕀 j:𝕀(i=1)(j=1)\Lambda_0^2 \coloneqq \sum_{i:\mathbb{I}} \sum_{j:\mathbb{I}} (i = 1) \vee (j = 1)

The walking cospan

The walking cospan or (2,0)-horn category is the category Λ 0 2\Lambda_0^2 which consists of three objects 0,1,2Λ 0 20, 1, 2 \in \Lambda_0^2 and two morphisms h 02:hom Λ 0 2(0,2)h_{02}:\mathrm{hom}_{\Lambda_0^2}(0, 2) and h 12:hom Λ 0 2(1,2)h_{12}:\mathrm{hom}_{\Lambda_0^2}(1, 2).

The category Λ 0 2\Lambda_0^2 is also called the (2,0)-horn preorder or the (2,0)-horn poset because Λ 0 2\Lambda_0^2 can be shown to be a poset.

Every cospan in a category CC is represented by a functor F:Λ 0 2CF:\Lambda_0^2 \to C from the walking cospan to CC.

In type theoretic approaches to synthetic (,1)(\infty,1)-category theory, the (2,0)-horn type Λ 0 2\Lambda_0^2 defined above plays the role of the walking cospan.

Properties

Right quotients

Following the notion of a right quotient of two elements in a monoid, we can define in category the notion of a right quotient of a cospan in a category.

A right quotient of a cospan g:hom A(y,z)g:\mathrm{hom}_A(y, z) and h:hom A(x,z)h:\mathrm{hom}_A(x, z) is a morphism f:hom A(x,y)f:\mathrm{hom}_A(x, y) such that hh is the unique composite of ff and gg, gf=hg \circ f = h.

Given a morphism f:hom A(x,y)f:\mathrm{hom}_A(x, y), the cospan (f,id y)(f, \mathrm{id}_y) is always right divisible with right quotient ff. A section is a right quotient of the span (id y,f)(\mathrm{id}_y, f).

The bicategory of cospans

Cospans in a category VV with small colimits form a bicategory whose objects are the objects of VV, whose morphisms are cospans between two objects, and whose 2-morphisms η\eta are commuting diagrams of the form

S σ S τ S a η b σ T τ T T. \array{ && S \\ & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S} \\ a &&\downarrow^\eta&& b \\ & {}_{\sigma_T}\searrow && \swarrow_{\tau_T} \\ && T } \,.

The category of cospans from aa to bb is naturally a category enriched in VV: for

S σ S τ S a b σ T τ T T \array{ && S \\ & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S} \\ a &&&& b \\ & {}_{\sigma_T}\searrow && \swarrow_{\tau_T} \\ && T }

two parallel cospans in VV, the VV-object a[S,T] b{}_a[S,T]_b of morphisms between them is the pullback

a[S,T] b pt σ T×τ T [S,T] σ S *×σ T * [ab,T] \array{ {}_a[S,T]_b &\to& pt \\ \downarrow && \downarrow^{\sigma_T \times \tau_T} \\ [S,T] &\stackrel{\sigma_S^* \times \sigma_T^*}{\to}& [a \sqcup b, T] }

formed in analogy to the enriched hom of pointed objects. The initial object of VV is the coproduct of aa and bb.

If VV has a terminal object, ptpt, then cospans from ptpt to itself are bi-pointed objects in VV.

References

Topological cospans and their role as models for cobordisms are discussed in

  • Marco Grandis, Collared cospans, cohomotopy and TQFT (Cospans in algebraic topology, II) (pdf)

Last revised on June 1, 2025 at 11:44:27. See the history of this page for a list of all contributions to it.