Is the ultraproduct concept fundamentally category-theoretic?

Question

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.

My question is whether the ultraproduct construction is fundamentally a category-theoretic concept.

The ultraproduct/ultrapower construction of Łos is used pervasively in logic, particularly in model theory and also in set theory, where nearly all of the larger large cardinal axioms can be formulated in terms of the existence of certain kinds of ultrapowers of the universe.

My question is, is the ultraproduct fundamentally a category-theoretic construction, in the sense that it is characterized by some natural category-theoretic universal property? How about the special case of ultrapowers?

I would be very interested, if there were a natural universal characterization in terms of the usual Hom sets for these first order structures, namely, first order elementary embeddings and/or homomorphisms. (Needless to say, I would be much less interested in a characterization that amounted merely to a translation of the Łos construction or of Łos's theorem into category-theoretic language.)

Background. Suppose we have a collection of structures $M_i$ for $i \in J$, all of the same first order type (e.g. groups, partial orders, graphs, fields, whatever), and $U$ is an ultrafilter on the index set $J$. This means that $U$ is a nonempty collection of nonempty subsets of $J$, containing every set or its complement, and closed under intersection and superset. The ultraproduct $\prod M_i / U$ consists of equivalence classes $[f]_U$, where $f$ is a function with domain $J$, with $f(i) \in M_i$, and $f \sim_U g$ iff $\{i \in J\; |\; f(i)=g(i)\} \in U$. One imposes structure on the ultraproduct by saying that a relation holds in the product, if it holds on a set in $U$, and similarly for functions. Łos's theorem then states that the ultraproduct satisfies a first order formula $\varphi([f]_U)$ if and only if $\{i \in J\; |\; M_i \text{ satisfies } \varphi(f(i))\}$ is in $U$. That is, truth in the ultraproduct amounts to truth on a $U$-large set of coordinates. The special case when all $M_i$ are the same model $M$, we arrive at the ultrapower $M^J /U$. In this case, there is a natural map from $M$ into $M^J / U$, defined by $x$ maps to $[c_x]_U$, where $c_x$ is the constant function with value $x$. It is easy to see that this map is an elementary embedding from $M$ into the ultrapower.

This question is a more focused instance of a probably-too-general question I asked here, and I may have several more in the future.

Answer 1

Yes. This is the content of the final section of my paper Codensity and the ultrafilter monad. Loosely, what's shown there is:

There is a standard piece of categorical machinery which, when fed as input the concept of finiteness of a family of structures, produces as output the concept of ultraproduct.

Let me say immediately that the result is due not to me, but to the anonymous referee. In the version of the paper I submitted to the journal (and in earlier arXiv versions), the final section essentially said "it looks like we should be able to describe the ultraproduct construction as a codensity monad, but I don't see how". The referee showed how, and I've included his or her theorem in the final version of the paper.

The "standard piece of categorical machinery" is the notion of codensity monad. "Recall" that (subject to the existence of certain limits) any functor $G\colon \mathcal{B} \to \mathcal{A}$ induces a monad on $\mathcal{A}$, the codensity monad of $G$. In the case where $G$ has a left adjoint $F$, this is just the monad $GF$, but codensity monads are defined in much wider generality.

(If you don't know what a monad is, then for the purposes of this answer, you can interpret "monad on $\mathcal{A}$" as "functor $\mathcal{A} \to \mathcal{A}$", although it's rather more than that.)

Fix a category $\mathcal{E}$ with small products and filtered colimits. (In model theory, this might typically be the category of structures for some finitary signature.) Let $\mathbf{Fam}(\mathcal{E})$ be the category in which an object is a set $X$ together with a family $(S_x)_{x \in X}$ of objects $S_x$ of $\mathcal{E}$. I'll skip the definition of the maps, but you can find it in my paper.

The ultraproduct construction determines a monad on $\mathcal{E}$, as follows. Given a family $$ S = (S_x)_{x \in X} $$ of objects of $\mathcal{E}$, taking ultraproducts produces a new family $$ \Bigl( \prod\nolimits_{\mathcal{U}} S \Bigr)_{\text{ultrafilters } \mathcal{U} \text{ on } X} $$ of objects of $\mathcal{E}$, where $\prod\nolimits_{\mathcal{U}} S$ denotes the ultraproduct of $(S_x)_{x \in X}$ with respect to $\mathcal{U}$. So, it's plausible that the ultraproduct construction gives at least a functor $\mathbf{Fam}(\mathcal{E}) \to \mathbf{Fam}(\mathcal{E})$. In fact, it gives not just a functor but a monad on $\mathbf{Fam}(\mathcal{E})$, the ultraproduct monad for $\mathcal{E}$.

The theorem is that this is a codensity monad. Specifically, let $\mathbf{FinFam}(\mathcal{E})$ be the full subcategory of $\mathbf{Fam}(\mathcal{E})$ consisting of those objects $(S_x)_{x \in X}$ in which the indexing set $X$ is finite. Then:

Theorem The codensity monad of the inclusion functor $\mathbf{FinFam}(\mathcal{E}) \hookrightarrow \mathbf{Fam}(\mathcal{E})$ is the ultraproduct monad for $\mathcal{E}$.

Notice that the concept of ultrafilter isn't taken as given. Actually, the concept of ultrafilter also arises as a codensity monad. This is a 1971 theorem of Kennison and Gildenhuys (discussed in my paper):

Theorem (Kennison and Gildenhuys) The codensity monad of the inclusion functor $\mathbf{FinSet} \hookrightarrow \mathbf{Set}$ is the ultrafilter monad.

Here $\mathbf{FinSet}$ is the category of finite sets, and the ultrafilter monad is the monad on $\mathbf{Set}$ that sends a set $X$ to the set of ultrafilters on $X$.

Answer 2

Andrej's answer was very helpful to me, but there is yet another (perhaps not completely unrelated) category theoretic view of ultraproducts that I am aware of. I am still hopeful that more category theorists will eventually step in and sort things out...

If $X$ is a discrete space then a sheaf $F:O(X)^{\mathrm{op}}\to Set$ must be such that $F(A) \cong \prod_{i \in A} F_i$ for some family of sets $(F_i)_{i \in X}$. This sheaf can be moved to a sheaf $F':O(\beta X)^{\mathrm{op}}\to Set$. Viewing $\beta X$ as the space of ultrafilters on $X$, the stalk of $F'$ at a point $\mathcal{U} \in \beta X$ is precisely the ultraproduct $\prod_{i \in X} F_i/\mathcal{U}$. (There is one subtle difference which occurs when some of the components $F_i$ are empty, in which case this ultraproduct can still be nonempty when $F(A)$ is nonempty for some $A \in \mathcal{U}$.)

From a more global point of view, the embedding $X \to \beta X$ induces a geometric morphism $Sh(X) \to Sh(\beta X)$. Similarly, a point of $\beta X$ can be identified with geometric morphism $\mathcal{U}:Set \to Sh(\beta X)$. The corresponding ultraproduct map is simply the composition $$Sh(X) \to Sh(\beta X) \xrightarrow{\mathcal{U}^*} Set,$$ where the last component is the inverse image part of $\mathcal{U}$. The corresponding ultrapower functor is the composite $$Set \xrightarrow{\Delta} Sh(X) \to Sh(\beta X) \xrightarrow{\mathcal{U}^*} Set,$$ where $\Delta$ is the diagonal functor.

Of course, there is nothing very special about discrete spaces in the above construction. The same construction exists for any completely regular Hausdorff space $X$ or, more generally, for a completely regular locale. (This makes sense even when the space/locale $X$ is not completely regular, but the map $X \to \beta X$ is not necessarily an embedding.) Of course, Łoś's Theorem takes a different form for this more general construction, the correct form of the theorem for a space/locale X can be found via the Kripke-Joyal semantics, for example.

Answer 3

The other answers so far have generally taken ultrafilters as a given, or used the Stone-Cech compactification (which has a universal property in Top). I'd like to point out that the set of ultrafilters on a set $I$ has a categorical interpretation in Sets. In particular, consider the diagram in Sets consisting of finite partitions $X_i\subset 2^I$ of $I$, with an arrow $X_i\to X_j$ if $X_i$ is a refinement of $X_j$; the arrow sends a subset of $I$ in $X_i$ to the unique element of $X_j$ containing it. Then the set of ultrafilters on $I$ is the inverse limit of this diagram.

Answer 4

This paper comes to mind:

Ultrasheaves and double negation. S. Awodey and J. Eliasson, Notre Dame Journal of Formal Logic 45(4), pp. 235--245 (2004). Available at http://www.andrew.cmu.edu/user/awodey/preprints/udn.pdf

Perhaps this is not quite what you are asking for, because the paper takes utrafilters as given, but it certainly gives a useful category-theoretic perspective.

Answer 5

For a short and painless description of how ultraproducts are colimits see page 6 of this article (it is equivalent, but I found it more readable than the descriptions given in the answers):

H. Mariano, F. Miraglia, Profinite structures are retracts of ultraproducts of finite structures, Reports on Mathematical Logic 42 (2007) pp169-182 (abstract, pdf)

The article contains a nice application of this categorical description of ultraproducts...

Answer 6

The notion of an ultrapower and more generally reduced powers and their generalizations are essentially category theoretic. More specifically, reduced powers are essentially pro-sets. This answer is a part of my own research, but these results are not ready to publish yet. Although these results relate the ultrapower construction to categories, I do not see how these results could generalize to possibly relate ultraproducts to categories.

In the other answers to this question, people have explained how ultraproducts are direct limits. It turns out that reduced powers are directed limits as well. Furthermore, I will show that as categories, the category of generalized reduced powers is equivalent to the category of inverse systems of sets. By the category of generalized reduced powers, I mean the category of most model-theoretic generalizations of the ultrapower and reduced power construction such as extenders, iterated ultrapowers, limit ultrapowers 3, Boolean ultrapowers 4, and their reduced power analogues such as limit reduced powers.

If $\mathcal{C}$ is a category, then the category $\mathbf{Pro}(\mathcal{C})$ is essentially the category of inverse systems over the category $\mathcal{C}$ and it should be thought of as the category of inverse limits from the category $\mathcal{C}$. I claim that the category $\mathbf{Pro}(\mathbf{Set})$ of pro-sets is equivalent to a full subcategory of pro-filters. Furthermore, these pro-filters are the things that we want to construct reduced powers. In fact, we obtain a three way duality between the category of pro-sets, the full subcategory of pro-filters where the transitional mappings are epimorphisms, and categories of reduced powers of structures.

$\large\mathbf{Categories}$ In this section, I will first define the categories and I will state the equivalences between these categories.

Let $A$ be a fixed infinite set. Let $\Omega(A)$ be the algebra with universe $A$ and where every operation is a fundamental operation. Then every object in $V(\Omega(A))$ is isomorphic to a limit reduced power of $\Omega(A)$ and any elementary extension of $\Omega(A)$ is isomorphic to a limit ultrapower of $\Omega(A)$ (see 3 for information on limit ultrapowers). Also, one may represent algebras in $V(\Omega(A))$ as limit reduced powers in such a way so that the homomorphisms between algebras in $V(\Omega(A))$ are induced by mappings between the underlying sets of the limit reduced powers. Since every algebra in $V(\Omega(A))$ is isomorphic to a limit reduced power of of $\Omega(A)$, one should think of the algebras in $V(\Omega(A))$ as limit reduced powers.

A small nonempty category $D$ is said to be a cofiltrant category if whenever $e_{1},e_{2}\in D$, then there is some $d\in D$ and morphisms $\ell_{1}:d\rightarrow e_{1},\ell_{2}:d\rightarrow e_{2}$, and if $\ell_{1},\ell_{2}:d\rightarrow e$ are morphisms, then there is some object $c$ and a morphism $m:c\rightarrow d$ with $\ell_{1}m=\ell_{2}m$. Every downward directed set is a cofiltrant category, and the notion of a cofiltrant category is the categorization of the notion of a downward directed set.

If $\mathcal{C}$ is a category, then $\mathbf{Pro}(\mathcal{C})$ is the category of all functors $F:\mathcal{D}\rightarrow\mathcal{C}$ for some cofiltrant category $\mathcal{D}$. One should think of the category $\mathbf{Pro}(\mathcal{C})$ as the category of all inverse limits in $\mathcal{C}$ since the notion of a cofiltrant category is the categorization of the notion of a downward directed set.

If $F:\mathcal{D}\rightarrow\mathcal{C},G:\mathcal{E}\rightarrow\mathcal{C}$ are objects in $\mathbf{Pro}(\mathcal{C})$, then define the set of morphisms by $$\mathrm{Hom}(F,G)=\varprojlim_{e\in\mathcal{E}}\ \varinjlim_{d\in\mathcal{D}}\ \mathrm{Hom}(F(d),G(e)).$$ The transitional mappings in the direct and inverse limits are the canonical ones, and the definition of composition of morphisms is defined in the natural way.

The following categories are the object of study in the papers 1 and 2.

We shall now construct a category $\mathfrak{F}$. The objects in $\mathfrak{F}$ are pairs $(X,\mathcal{F})$ where $X$ is a set and $\mathcal{F}$ is a filter on $X$. If $(X,\mathcal{F}),(Y,\mathcal{G})\in\mathfrak{F}$, then function $f:X\rightarrow Y$ is a morphism from $(X,\mathcal{F}),(Y,\mathcal{G})$ if $f^{-1}[R]\in\mathcal{F}$ whenever $R\in\mathcal{G}$. It is easy to show that $f$ is a morphism from $(X,\mathcal{F})$ to $(Y,\mathcal{G})$ if and only if whenever $R\subseteq X$ is non-negligible (i.e. $R^{c}\not\in\mathcal{F}$), then the image $f[R]$ is non-negligible. The intuition behind defining the category $\mathfrak{F}$ this way is that we do not want to map non-negligible sets to negligible sets since that is like a function mapping a non-empty set to an empty set.

Let $\mathfrak{G}$ be the quotient category of $\mathfrak{F}$ where we relate two morphisms if they differ by a negligible set. In other words, the objects in $\mathfrak{G}$ and $\mathfrak{F}$ are the same. If $(X,\mathcal{F}),(Y,\mathcal{G})$ are objects in $\mathfrak{F}$, and $f,g\in\mathrm{Hom}_{\mathfrak{F}}[(X,\mathcal{F}),(Y,\mathcal{G})]$, then $f\simeq g$ iff $\{x\in X|f(x)=g(x)\}\in\mathcal{F}$. Then $\mathrm{Hom}_{\mathcal{G}}[(X,\mathcal{F}),(Y,\mathcal{G})]=\mathrm{Hom}_{\mathfrak{F}}[(X,\mathcal{F}),(Y,\mathcal{G})]/\simeq$. The composition in $\mathfrak{G}$ is defined in the obvious manner.

While the categories $\mathfrak{F}$ and $\mathfrak{G}$ were both studied in 1 and 2, the category $\mathfrak{G}$ is more fundamental than $\mathfrak{F}$ and the category $\mathfrak{G}$ deserves to be called the category of filters while $\mathfrak{F}$ does not seem to be very useful.

Morphisms between algebras in $\mathfrak{G}$ induce homomorphisms between reduced powers. If $\mathcal{A}$ is an algebraic structure and $[f]:(X,\mathcal{F})\rightarrow(X,\mathcal{G})$ is a morphism in $\mathfrak{G}$, then we define a morphism $[f]^{\bullet}:\mathcal{A}^{X}/\mathcal{G}\rightarrow\mathcal{A}^{Y}/\mathcal{F}$ by letting $[f]^{\bullet}[\ell]=[\ell\circ f]$.

$\mathbf{Proposition}$ Let $[f]\in \mathrm{Hom}_{\mathfrak{G}}[(X,\mathcal{F}),(Y,\mathcal{G})]$. The following are equivalent.

1. If $[f]$ is an epimorphism in $\mathfrak{G}$.

2. $\mathcal{G}=\{S\subseteq Y|f^{-1}[S]\in\mathcal{F}\}$

3. If $R\in\mathcal{F}$, then $f[R]\in\mathcal{G}$.

4. The canonical mapping $[f]^{\bullet}:A^{Y}/\mathcal{G}\rightarrow A^{X}/\mathcal{F}$ is injective for each set $A$.

Let $\mathbf{PF}$ be the full subcategory of $\mathbf{Pro}(\mathfrak{G})$ whose objects consist of inverse systems $((X_{d},\mathcal{F}_{d})_{d\in D},(\ell_{d_{1},d_{2}})_{d_{1}\leq d_{2}})$ over directed sets $D$ such that each $\ell_{d_{1},d_{2}}$ is an epimorphism in $\mathfrak{G}$.

$\mathbf{Humor}:$ I call the objects in $\mathbf{PF}$ pro-filters. That is, professional filters. The pro-filters play football in the National Filter League (NFL).

If $\kappa$ is a cardinal, then let $\mathbf{PF}_{\kappa}$ denote the full subcategory of $\mathbf{PF}$ consisting of inverse systems $(X_{d},\mathcal{F}_{d})_{d\in D}$ such that $|X_{d}|<\kappa$ for each $d\in D$. Let $\mathbf{Set}_{\kappa}$ denote the category of sets of cardinality less than $\kappa$. Let $\mathbf{U}_{\kappa}$ be the subcategory of $\mathbf{PF}_{\kappa}$ consisting of all inverse systems $(X_{d},\mathcal{F}_{d})_{d\in D}$ where each $\mathcal{F}_{d}$ is an ultrafilter.

$\mathbf{Theorem}$ Let $A$ be an infinite set.

1. The category $\mathbf{PF}$ is covariantly equivalent to the category $\mathbf{Pro}(\mathbf{Set})$.

2. The functor defined by $(X_{d},\mathcal{F}_{d})_{d\in D}\mapsto \Omega(A)^{X_{d}}/\mathcal{F}_{d}$ is a contravariant equivalence between the category $\mathbf{PF}_{|A|^{+}}$ and the category $V(\Omega(A))$. Furthermore, this functor restricts to an equivalence between the category $\mathbf{Pro}(\mathbf{U}_{|A|^{+}})$ and the elementary extensions of $\Omega(A)$.

3. The category $\mathbf{Pro}(\mathbf{Set}_{|A|^{+}})$ is contravariantly equivalent to the category $V(\Omega(A))$.

The equivalences between the category $\mathbf{PF}$ and $\mathbf{Pro}(\mathbf{Set})$ can be described explicitly. If $((X_{d},\mathcal{F}_{d})_{d\in D},(\ell_{d_{1},d_{2}})_{d_{1}\leq d_{2}})\in\mathbf{PF}$, then add a transitional mapping $f:X_{d}\rightarrow X_{e}$ whenever $f\in\textrm{Hom}_{\mathfrak{F}}((X_{d},\mathcal{F}_{d}),(X_{e},\mathcal{F}_{e}))$ and $[f]=\ell_{d,e}$.

If $F:\mathcal{D}\rightarrow\mathbf{Set}$ is a pro-set, then for each $d\in\mathcal{D}$, then $\{\mathrm{Im}(F(f))|f:e\rightarrow d\,\textrm{for some}\,e\in\mathcal{D}\}$ is a filterbase that generates a filter on $F(d)$ which we shall denote by $\mathcal{F}_{d}$. Partial order $\mathcal{D}$ where $d\leq e$ iff there is a morphism from $d$ to $e$. Then if $d\leq e$, then let $\ell_{d,e}:(X_{d},\mathcal{F}_{d})\rightarrow(X_{e},\mathcal{G}_{e})$ be the morphism where $\ell_{d,e}=[F(f)]$ for some $f:d\rightarrow e$. It is easy to show that the morphism $\ell_{d,e}$ does not depend on $[F(f)]$ and $((X_{d},\mathcal{F}_{d}),(\ell_{d,e})_{d\leq e})\in\mathbf{PF}$.

$\mathbf{Corollary}$ If $|A|<|B|$, then the category $V(\Omega(A))$ is equivalent to a full subcategory of $V(\Omega(B))$.

$\mathbf{Corollary}$ If $(X_{d},\mathcal{F}_{d})_{d\in D},(Y_{e},\mathcal{G}_{e})_{e\in E}\in\mathbf{PF}_{|A|^{+}}$, and the algebras $^{\lim}_{\longrightarrow}\Omega(A)^{X_{d}}/\mathcal{F}_{d}$ and $^{\lim}_{\longrightarrow}\Omega(A)^{Y_{e}}/\mathcal{G}_{e}$ are isomorphic, then $^{\lim}_{\longrightarrow}\mathcal{A}^{X_{d}}/\mathcal{F}_{d}$ and $^{\lim}_{\longrightarrow}\mathcal{A}^{Y_{e}}/\mathcal{G}_{e}$ are isomorphic for each structure $\mathcal{A}$.

The above result still holds when one replaces the direct limit of reduced powers with other reduced power and ultrapower constructions.

$\large\textbf{An Application}$

We shall now give an application that shows that going between algebras and different categories may be useful. The following result can be proved using the duality between pro-filters and algebras (the proof also uses a version of the Feferman-Vaught theorem, and the Keisler-Shelah isomorphism theorem or Frayne's theorem).

$\mathbf{Theorem}$ Let $\mathbf{2}$ denote the two element Boolean algebra. Let $\mathcal{A}\mapsto\mathbf{R}(\mathcal{A}),\mathcal{A}\mapsto\mathbf{S}(\mathcal{A})$ be two distinct reduced power constructions(such as limit reduced powers, Boolean powers etc.). If $\mathbf{R}(\mathbf{2})$ and $\mathbf{S}(\mathbf{2})$ are elementarily equivalent, then there is a sequence of ultrafilters $\mathcal{U}_{n}$ such that for every structure $\mathcal{A}$, we have $$^{\lim}_{n\rightarrow\infty}(\mathbf{R}(\mathcal{A}))^{\mathcal{U}_{0}\cdots\mathcal{U}_{n}}\simeq^{\lim}_{n\rightarrow\infty}(\mathbf{S}(\mathcal{A}))^{\mathcal{U}_{0}\cdots\mathcal{U}_{n}}.$$

We take note that in the above result we cannot replace the direct limit of ultrapowers with a single ultrapower. For example, if $\mathcal{U}$ is a non-principal ultrafilter, and $\mathbf{R}(\mathcal{A})=\mathcal{A}$ and $\mathbf{S}(\mathcal{A})=\mathcal{A}^{\mathcal{U}}$ for all structures $\mathcal{A}$, then for each non-principal ultrafilter $\mathcal{V}$, we have $\mathcal{V}<_{RK}\mathcal{U}\cdot\mathcal{V}$. In particular, the ultrafilters $\mathcal{V}$ and $\mathcal{U}\cdot\mathcal{V}$ are not Rudin-Kiesler equivalent, so there is some structure $\mathcal{A}$ where the structures $\mathcal{A}^{\mathcal{V}}$ and $\mathcal{A}^{\mathcal{U}\cdot\mathcal{V}}\simeq(\mathcal{A}^{\mathcal{U}})^{\mathcal{V}}$ are not isomorphic.

I must also mention that the elementary classes of Boolean algebras have a particularly nice and simple classification. There is a countable set of Boolean algebra invariants called the elementary invariants, and two Boolean algebras are elementarily equivalent if and only if they satisfy the same elementary invariants. In particular, to check that $\mathbf{R}(2)$ and $\mathbf{S}(2)$ are elementarily equivalent, it suffices to show that these Boolean algebras have the same elementary invariants. The reader is referred to [5] or [6][Vol. 1] for more information on these elementary invariants.

Oh the joy of cats!!!

1 Blass, Andreas Two closed categories of filters. Fund. Math. 94 (1977), no. 2, 129–143.

2 Koubek, Vaclav; Reiterman, Jan On the category of filters. Comment. Math. Univ. Carolinae 11 1970 19–29

3 Keisler, H. Jerome. Limit ultrapowers. Trans. Amer. Math. Soc. 107 1963 382–408

4 Mansfield, Richard. The theory of Boolean ultrapowers. Ann. Math. Logic 2 (1970/71), no. 3, 297–323.

[5] Chang, Chen Chung, and H. Jerome Keisler. Model Theory. Amsterdam: North-Holland Pub., 1973.

[6] Monk, J. Donald, Robert Bonnet, and Sabine Koppelberg. Handbook of Boolean Algebras. Amsterdam: North-Holland, 1989.

Answer 7

There are already several answers describing how ultraproducts arise naturally from categorical considerations. However, I don't think any of them address this part of the question:

I would be very interested, if there were a natural universal characterization in terms of the usual Hom sets

In fact, there is a universal characterisation of ultraproducts in any category, which looks similar to the characterisation of the universal properties of colimits in terms of universal cones. This is described in §5 of Tendas's Flatness, weakly lex colimits, and free exact completions.

Definition 5.1 (Tendas). Let $\mathcal M$ be a category and $\mathcal U$ be an ultrafilter on a set $X$. Given an $X$-indexed family of objects $(M_x)_{x\in X}$, the universal ultraproduct of $(M_x)_{x\in X}$ over $\mathcal U$ is an object $\textstyle\prod_{\mathcal U} M_x \in \mathcal M$ together with a cocone $$(\delta_S \colon \prod_{s \in S}\mathcal M(-, M_s) \to \mathcal M(-, \textstyle\prod_{\mathcal U} M_x))_{S \in \mathcal U}$$ over $\mathcal U$ which is universal among all other cocones with representable codomain.

When $\mathcal M$ has products, universal ultraproducts coincide with the categorical ultraproducts of Lurie's Ultracategories (Proposition 5.4 of Tendas).

Answer 8

Let $\Phi$ a filter on the set $I$ and let $X_i \in C\ i\in I$, and for $U \subset I$ let $X_U:=\prod_{i\in U} X_i$ . We have a natural functor $T : P(I)^{op} \to C$ (where $P(I)$ the order of ubset onf $I$) as: $T(U):=X_U$ and for $T(V\subset U) : X_U \to X_V$ the canonical proiection induced by $V \subset U$. Being $I$ the initial object of $P(I)$ and letting $X:=X_I$ follow a "lifting" of $T$ to a functor $T': P(I)^{op} \to X \downarrow C$ as $T'(U)= (T(U\subset I), X_U)$ and $T'(V\subset U)=T(V\subset U) $. For any $U \in \Phi$ let $\alpha_U, \beta_U : K_U \to X$ the kernel pair of $T'(U)$ we obtain a diagram of all these $\alpha_U, \beta_U$ morphisms and the natural $K_{V \subset U}: K_U\to K_V$ induced by $T(V \subset U)$. The colimit of this diagram is the ultrapower of $\prod_{i\in I} X_i$ respecto to $\Phi$. This colimit is the colimits in $X\downarrow C$ of the cokers of the kernel pairs.

But things are more simple too: observe that choise a retraction $r: I\to U$ os the inclusion $U\subset I$ we have a section $T(r): T(U)\to X$ on $T(U\subset I)$ give by $\pi_i\circ T(r) = \pi_{r(i)}$ then $T(U\subset I)$ being a retraction is a regular epimorphism (a coker of some pair), then is (well knowed fact) the coker of its Ker-pair $\alpha_U, \beta_U$. Then follow that the ultrapower is the colimit in $X\downarrow C$ of the diagram of $T(U\subset I)\ U\subset I$ as objects and $T(V\subset U)$ as morphisms.