nLab categories with finite products are cosifted



Limits and colimits

Category theory



A basic example of limits commuting with colimits in category theory is that colimits over opposite categories of categories with finite products preserves finite products. One says equivalently that categories with finite products are cosifted categories.



(categories with finite products are cosifted)

Let 𝒞\mathcal{C} be a small category which has finite products. Then 𝒞\mathcal{C} is a cosifted category, equivalently its opposite category 𝒞 op\mathcal{C}^{op} is a sifted category, equivalently colimits over 𝒞 op\mathcal{C}^{op} with values in Set are sifted colimits, equivalently colimits over 𝒞 op\mathcal{C}^{op} with values in Set commute with finite products, as follows:

For X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] to functors on the opposite category of 𝒞\mathcal{C} (hence two presheaves on 𝒞\mathcal{C}) we have a natural isomorphism

lim𝒞 op(X×Y)(lim𝒞 opX)×(lim𝒞 opY). \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) \;\simeq\; \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \,.

First observe that for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] two presheaves, their Cartesian product is a colimit over presheaves represented by Cartesian products in 𝒞\mathcal{C}. Explicity, using coend-notation, we have:

(1)X×Y c 1,c 2𝒞y(c 1×c 2)×X(c 1)×Y(c 2), \mathbf{X} \times \mathbf{Y} \;\simeq\; \int^{c_1,c_2 \in \mathcal{C}} y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \,,

where y:𝒞[𝒞 op,Set]y \;\colon\; \mathcal{C} \hookrightarrow [\mathcal{C}^{op}, Set] denotes the Yoneda embedding.

This is due to the following sequence of natural isomorphisms:

(X×Y)(c) ( c 1𝒞𝒞(c,c 1)×X(c 1))×( c 2𝒞𝒞(c,c 2)×Y(c 2)) c 1𝒞 c 2𝒞𝒞(c,c 1)×𝒞(c,c 2)𝒸(c,c 1×c 2)×(X(c 1)×X(c 2)) c 1𝒞 c 2𝒞𝒞(c,c 1×c 2)×X(c 1)×X(c 2), \begin{aligned} (\mathbf{X} \times \mathbf{Y})(c) & \simeq \left( \int^{c_1 \in \mathcal{C}} \mathcal{C}(c,c_1) \times \mathbf{X}(c_1) \right) \times \left( \int^{c_2 \in \mathcal{C}} \mathcal{C}(c,c_2) \times \mathbf{Y}(c_2) \right) \\ & \simeq \int^{c_1 \in \mathcal{C}} \int^{c_2 \in \mathcal{C}} \underset{ \simeq \mathcal{c}(c, c_1 \times c_2) }{ \underbrace{ \mathcal{C}(c,c_1) \times \mathcal{C}(c,c_2) }} \times \left( \mathbf{X}(c_1) \times \mathbf{X}(c_2) \right) \\ & \simeq \int^{c_1 \in \mathcal{C}} \int^{c_2 \in \mathcal{C}} \mathcal{C}(c,c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{X}(c_2) \,, \end{aligned}

where the first step expands out both presheaves as colimits of representables separately, via the co-Yoneda lemma, the second step uses that the Cartesian product of presheaves is a two-variable left adjoint (by the symmetric closed monoidal structure on presheaves) and as such preserves colimits (in particular coends) in each variable separately, and under the brace we use the defining universal property of the Cartesian products, assumed to exist in 𝒞\mathcal{C}.

Now observe that the colimit of a representable presheaf is the singleton.

(2)lim𝒟 opy(c)*. \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c) \;\simeq\; \ast \,.

One way to see this is to regard the colimit as the left Kan extension along the unique functor 𝒞 opp*\mathcal{C}^{op} \overset{p}{\to} \ast to the terminal category. By the formula there, this yields

lim𝒟 opy(c) c 1𝒞*(,p(c 1))const *(c 1)×y(c)(c 1) c 1𝒞const *(c 1)×𝒞(c 1,c) const *(c) * \begin{aligned} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c) & \simeq \int^{c_1 \in \mathcal{C}} \underset{const_\ast(c_1)}{\underbrace{\ast(-,p(c_1))}} \times y(c)(c_1) \\ & \simeq \int^{c_1 \in \mathcal{C}} const_\ast(c_1) \times \mathcal{C}(c_1,c) \\ & \simeq const_\ast(c) \\ & \simeq \ast \end{aligned}

where we made explicit the constant functor const *:𝒞Setconst_\ast \;\colon\; \mathcal{C} \to Set, constant on the singleton set *\ast, and then applied the co-Yoneda lemma.

Using this, we compute for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] the following sequence of natural isomorphisms:

lim𝒟 op(X×Y) lim𝒟 op c 1,c 2𝒞y(c 1×c 2)×X(c 1)×Y(c 2) c 1,c 2𝒞lim𝒟 op(y(c 1×c 2)×X(c 1)×Y(c 2)) c 1,c 2𝒞(lim𝒟 opy(c 1×c 2)*)×X(c 1)×Y(c 2) c 1,c 2𝒞(X(c 1)×Y(c 2)) ( c 1𝒞X(c 1))×( c 2𝒞Y(c 2)) (lim𝒞 opX)×(lim𝒞 opY) \begin{aligned} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) & \simeq \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \int^{c_1,c_2 \in \mathcal{C}} y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \left( y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \right) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \left( \underset{ \simeq \ast }{ \underbrace{ \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c_1 \times c_2) }} \right) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \left( \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \right) \\ & \simeq \left( \int^{c_1\in \mathcal{C}} \mathbf{X}(c_1) \right) \times \left( \int^{c_2\in \mathcal{C}} \mathbf{Y}(c_2) \right) \\ & \simeq \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \end{aligned}

Here the first step is (1), the second uses that colimits commute with colimits, the third uses again that the Cartesian product respects colimits in each variable separately, the fourth is (2), the last step is again the respect for colimits of the Cartesian product in each variable separately.


Last revised on December 30, 2023 at 11:57:45. See the history of this page for a list of all contributions to it.