For posetal categories with initial objects, the pushout-conservative cocompletion distributes over the product

Jul 8, 2025

This argument is also available as a PDF file, including some examples which are omitted here. The reader should have some background in category theory.

Thank you to Sam Staton and Owen Lynch for showing me a sketch proof of this result!

Preliminaries

Denote by 𝟚 the preorder {0 ≤ 1} of truth values, viewed as a category.

Note that if a category is a poset, then it is in particular a preorder; preorders are 𝟚-enriched categories [2].

For the rest of this section, let 𝒜 be an arbitrary finite posetal category with an initial object a₀, which we may view as a 𝟚-enriched category.

There is an isomorphism between downsets of 𝒜 and objects in the functor category [𝒜^op, 𝟚]. In particular, for any downset, define a functor by sending elements of the downset to 1 and the rest to 0; for any functor, define a downset by including all the objects sent to 1. Also note that arrows in this functor category are set inclusions between downsets. So the functor category is isomorphic to the posetal category of the set of all downsets of 𝒜.

Note that 𝟚 is a complete chain, and therefore is a completely distributive lattice; equalities using this fact will be marked =^†.

⋅ ← ⋅ → ⋅-colimits are joins and ⋅ → ⋅ ← ⋅-limits are meets in 𝒜

Since 𝒜 is thin, any triangle of arrows commutes.

So the colimit of a diagram y ← x → z does not depend on x; in fact, it is the join of y and z in the poset 𝒜.

Similarly, the limit of a digram b → a ← c does not depend on a; it is the meet of b and c in 𝒜.

Description of the pushout-conservative cocompletion of 𝒜

Kelly [3] gives a description for the R-conservative cocompletion of an arbitrary category, where R is some set of diagram schemes. We aim to make this concrete in the case of 𝒜 by using the description in [4]. Fix R = {⋅ ← ⋅ → ⋅} from this point onwards. We would like to find a concrete description for CPsh(𝒜), the pushout-conservative cocompletion of 𝒜. Note that since 𝒜 has an initial object, the condition of CPsh(𝒜) having all colimits is equivalent to the condition of it having all pushouts. So we may also view CPsh(𝒜) as the pushout-preserving pushout completion.

Description of [𝒜^op, 𝟚]_R^op

[𝒜^op, 𝟚]_R^op is the category of all functors 𝒜^op → 𝟚 which preserve ⋅ → ⋅ ← ⋅-limits. That is, these are functors sending ⋅ ← ⋅ → ⋅-colimits in 𝒜 to ⋅ → ⋅ ← ⋅-limits in 𝟚. But since both 𝒜 and 𝟚 are posets and by the description of limits/colimits of these diagram schemes, this is equivalent to functors which send joins in 𝒜 to meets in 𝟚. So the functors in [𝒜^op, 𝟚]_R^op are exactly those downsets that have an associated function F : ob(𝒜) → ob(𝟚) on objects satisfying if a ∨ b is defined, then F(a ∨ b) = F(a) ∧ F(b) (note that the only case in which this is a stronger condition than the condition of being a downset is when F(a) = F(b) = 1, in other words we additionally require that when both a and b are in the downset, so is their join).

Downsets generated by objects of 𝒜

A functor in [𝒜^op, 𝟚] generated by an object z of 𝒜 is the downset of all elements x such that x ≤ z; we denote such a functor/downset by ↓ z. Any such functor is also in [𝒜^op, 𝟚]_R^op since if any two elements a and b are in the downset, then they are comparable (by the existence of the bottom/initial element), and so their join is their maximum.

Closure under all small colimits

Finally, to obtain the pushout-conservative cocompletion of 𝒜, we close 𝒜 in [𝒜^op, 𝟚]_R^op under all (small) colimits. We aim to show that this closure consists of the subcategory of exactly those functors F that correspond to non-empty downsets satisfying [1].

The subcategory is closed under all non-empty colimits (arguing by contradiction, there is no inclusion arrow from any non-empty downset to the empty downset), and the colimit of the empty diagram is ↓ a₀, the initial object in the subcategory we have defined (recall that a₀ is the initial object of 𝒜).

The subcategory is also generated by representables, since any desired downset F = {x₁, x₂, ..., x_k} is the colimit of the diagram ( ↓ x₁ ↓ x₂ ... ↓ x_k ).

Description of CPsh(𝒜)

We conclude that CPsh(𝒜) consists of the poset of all non-empty downsets of 𝒜 which satisfy [1], ordered by the set inclusion relation.

Statement of the claim

Let 𝒞 and 𝒟 be finite posetal categories, each with an initial element. Then, their pushout-conservative cocompletions distribute over taking the product.

That is, CPsh(𝒞 × 𝒟) ≅ CPsh(𝒞) × CPsh(𝒟).

Proof of the claim

Define the functors G : CPsh(𝒞) × CPsh(𝒟) → CPsh(𝒞 × 𝒟) and H : CPsh(𝒞 × 𝒟) → CPsh(𝒞) × CPsh(𝒟) by G(F₁, F₂) = (p₁, p₂) ↦ F₁(p₁) ∧ F₂(p₂) , H(F) = (p₁ ↦ ⋁_{p₂ ∈ ob(𝒞)}F(p₁, p₂), p₂ ↦ ⋁_{p₁ ∈ ob(𝒟)}F(p₁, p₂)) .

For F₁ ∈ CPsh(𝒞) and F₂ ∈ CPsh(𝒟), we have G(F₁, F₂) ∈ CPsh(𝒞 × 𝒟)

Suppose we have (q₁, q₂) ≤ (p₁, p₂) in 𝒞 × 𝒟, and G(F₁, F₂)(p₁, p₂) = 1.

Then q₁ ≤ p₁ and q₂ ≤ p₂ and F₁(p₁) = F₂(p₂) = 1.

So F₁(q₁) = F₂(q₂) = 1 and G(F₁, F₂)(q₁, q₂) = 1.

So G(F₁, F₂) is a downset, and it’s clearly non-empty by the non-emptiness of each of F₁ and F₂.
If F₁ and F₂ satisfy [1], then G(F₁, F₂)((p₁, p₂) ∨ (q₁, q₂)) = G(F₁, F₂)(p₁ ∨ q₁, p₂ ∨ q₂) = F₁(p₁ ∨ q₁) ∧ F₂(p₂ ∨ q₂) = F₁(p₁) ∧ F₁(q₁) ∧ F₂(p₂) ∧ F₂(q₂) = G(F₁, F₂)(p₁, p₂) ∧ G(F₁, F₂)(q₁, q₂) and we conclude that G(F₁, F₂) satisfies [1].

For F ∈ CPsh(𝒞 × 𝒟), we have H(F) ∈ CPsh(𝒞) × CPsh(𝒟)

By symmetry it suffices to check that (p₁ ↦ ⋁_{p₂ ∈ ob(𝒞)}F(p₁, p₂)) ∈ CPsh(𝒞).

Suppose we have q₁ ≤ p₁ in 𝒞, and ⋁_{p₂ ∈ ob(𝒞)}F(p₁, p₂) = 1.

So ∃p₂ ∈ ob(𝒞) such that F(p₁, p₂) = 1.

Since (q₁, p₂) ≤ (p₁, p₂) in 𝒞 × 𝒟, we have F(q₁, p₂) = 1 and so ⋁_{p₂ ∈ ob(𝒞)}F(q₁, p₂) = 1.

Since F is non-empty, there is some (p′₁, p′₂) such that F(p′₁, p′₂) = 1. Then, taking p′₁ as input suffices to check the non-emptiness of (p₁ ↦ ⋁_{p₂ ∈ ob(𝒞)}F(p₁, p₂)).
If F satisfies [1], then ⋁_{p₂ ∈ ob(𝒞)}F(p₁ ∨ q₁, p₂) = ⋁_{p₂ ∈ ob(𝒞)}F((p₁, p₂) ∨ (q₁, p₂)) = ⋁_{p₂ ∈ ob(𝒞)}F(p₁, p₂) ∧ F(q₁, p₂) =^†⋁_{p₂ ∈ ob(𝒞)}F(p₁, p₂) ∧ ⋁_{p′₂ ∈ ob(𝒞)}F(q₁, p′₂) and we conclude that H(F) satisfies [1].

G ∘ H is identity on objects of CPsh(𝒞 × 𝒟)

We have shown above that CPsh(𝒞 × 𝒟) is posetal, so it suffices to show an inequality in each direction.

We have (G ∘ H)(F)(p₁, p₂) = ⋁_{p′₂ ∈ ob(𝒞)}F(p₁, p′₂) ∧ ⋁_{p′₁ ∈ ob(𝒟)}F(p′₁, p₂) =^†⋁_{p′₂ ∈ ob(𝒞)}⋁_{p′₁ ∈ ob(𝒟)}F(p₁, p′₂) ∧ F(p′₁, p₂) = max_{p′₂ ∈ ob(𝒞), p′₁ ∈ ob(𝒟)}F(p₁, p′₂) ∧ F(p′₁, p₂) (since 𝟚 is a total order) ≥ F(p₁, p₂) ∧ F(p₁, p₂) = F(p₁, p₂) and further, for an arbitrary p′₂ ∈ ob(𝒞) and p′₁ ∈ ob(𝒟) we have F(p₁, p′₂) ≤ F(p₁, ⊥) and F(p′₁, p₂) ≤ F(⊥, p₂) since F is a downset. So F(p₁, p′₂) ∧ F(p′₁, p₂) ≤ F(p₁, ⊥) ∧ F(⊥, p₂) ≤ F((p₁, ⊥) ∨ (⊥, p₂)) ≤ F(p₁, p₂) and we conclude that (G ∘ H)(F)(p₁, p₂) = max_{p′₂ ∈ ob(𝒞), p′₁ ∈ ob(𝒟)}(F(p₁, p′₂) ∧ F(p′₁, p₂)) ≤ F(p₁, p₂).

H ∘ G is identity on objects of CPsh(𝒞) × CPsh(𝒟)

By symmetry it suffices to check the first component of (H ∘ G)(F₁, F₂), call it ((H ∘ G)(F₁, F₂))₁. We have ((H ∘ G)(F₁, F₂))₁(p₁) = ⋁_{p₂ ∈ ob(𝒞)}F₁(p₁) ∧ F₂(p₂) =^†F₁(p₁) ∧ ⋁_{p₂ ∈ ob(𝒞)}F₂(p₂) = F₁(p₁) ∧ 1 (since F₂ is a non-empty downset) = F₁(p₁) as required.

G ∘ H and H ∘ G are identities on arrows

Since CPsh(𝒞 × 𝒟) and CPsh(𝒞) × CPsh(𝒟) are thin categories, this condition is trivially satisfied.

References

[2]: relation between preorders and (0,1)-categories in nLab. url: https://ncatlab.org/nlab/show/relation+between+preorders+and+%280%2C1%29-categories (visited on 07/04/2025).

[3]: Gregory Maxwell Kelly. Basic concepts of enriched category theory. Vol. 64. CUP Archive, 1982. isbn:0-521-28702-2.

[4]: Jiří Velebil and Jiří Adámek. “A remark on conservative cocompletions of categories”. In: Journal of Pure and Applied Algebra 168.1 (2002), pp. 107–124. issn: 0022-4049. doi: https://doi.org/10.1016/S0022-4049(01)00051-2. url: https://www.sciencedirect.com/science/article/pii/S0022404901000512.