Exploring 16 Types Of 2-Dimensional CoLimits In Bicategories
Hey guys! Ever feel like the world of category theory is a bit… one-dimensional? Well, buckle up, because we're about to dive headfirst into the wild and wonderful world of bicategories, where limits and colimits explode into a dazzling array of 16 different flavors! That's right, you heard me – sixteen! Forget your simple left and right, up and down; we're talking a full-blown 2-dimensional (co)limit fiesta!
The 1-Dimensional World: A Quick Recap of Limits and Colimits
Before we jump into the bicategorical deep end, let's do a quick refresher on the good ol' 1-dimensional world of categories. In this familiar setting, we have two fundamental notions of (unweighted) limits: limits and colimits. Think of limits as ways to find the “most general” object that satisfies certain conditions, while colimits do the opposite – they find the “most specific” object. For instance, the product of two objects is a limit, representing the most general object admitting maps to both. Dually, the coproduct is a colimit, representing the most specific object receiving maps from both. Limits capture notions of universal constructions, providing a powerful tool for constructing objects with desired properties.
In the realm of ordinary, garden-variety 1-categories, the concept of limits and colimits is relatively straightforward. You've got your basic limits like products, pullbacks, equalizers, and their dual colimit counterparts: coproducts, pushouts, coequalizers. These are the bread and butter of category theory, allowing us to construct new objects from existing ones in a structured and principled way. But, guys, what happens when we crank up the dimensionality dial to eleven? What happens when we venture into the land of bicategories, where the morphisms themselves have morphisms between them? That's where things get really interesting.
Why Bicategories Matter: A Sneak Peek into Higher Dimensions
So, what's the big deal about bicategories anyway? Why should we care about these seemingly esoteric structures? Well, bicategories are crucial for capturing situations where composition of morphisms is not strictly associative or unital, but only associative and unital up to isomorphism. This might sound like a technicality, but it turns out to be incredibly common in many areas of mathematics. Think of situations involving homotopies, natural transformations, or even the composition of functors. These are all examples where strict equality just doesn't cut it; we need the flexibility of working up to isomorphism.
Consider, for instance, the category of topological spaces and continuous maps. The composition of continuous maps is strictly associative. However, when we move to the homotopy category, where we consider continuous maps up to homotopy, this strict associativity breaks down. The composition of homotopy classes is only associative up to homotopy. Bicategories provide the perfect framework for dealing with these situations, allowing us to formalize and reason about higher-dimensional structures and relationships.
Enter the Bicategory: Where Limits Get a 2-Dimensional Upgrade
Now, let's talk bicategories. In a bicategory, we don't just have objects and morphisms; we also have 2-morphisms between morphisms. Think of it like this: you have objects, arrows between objects, and then arrows between arrows. This extra layer of structure gives bicategories a richness that 1-categories simply lack. It allows us to capture more nuanced relationships and constructions, particularly when it comes to limits and colimits.
The key difference between limits in 1-categories and bicategories lies in how we handle the universal property. In a 1-category, a limit is an object equipped with morphisms satisfying a universal property, meaning any other object with similar morphisms factors uniquely through the limit. In a bicategory, this notion gets weakened. Instead of requiring a unique factorization, we only require a factorization that is unique up to a 2-isomorphism. This seemingly small change has profound consequences, leading to the explosion of (co)limit flavors we mentioned earlier.
The Sixteen Faces of (Co)limits in Bicategories: A Taxonomy
Okay, guys, here's the main event: the sixteen different notions of (co)limits in bicategories. In the world of 1-categories, we have two fundamental options: limits and colimits. However, when we introduce the 2-dimensional structure of bicategories, this simple dichotomy fractures into a more complex landscape. The reason for this explosion of (co)limit concepts stems from the fact that we have choices to make about how strictly we want our (co)limits to behave. Do we want the universal property to hold up to isomorphism, or do we want something stronger? Do we care about the direction of the 2-morphisms involved? These questions lead to a plethora of different definitions, each capturing a slightly different aspect of the (co)limit concept.
Each of these notions has an adjective modifying (co)limit, which can be: lax, pseudo, or strict, in the variance in which the cone/cocone condition holds. They also modify comma objects, which are special cases of (co)limits. These adjectives describe how strictly the universal property of the (co)limit holds. A strict (co)limit satisfies the universal property on the nose, with equality. A pseudo (co)limit satisfies it up to isomorphism, and a lax (co)limit satisfies it up to a 2-morphism, but not necessarily an invertible one.
Further, we also have notions of icons and opicons which modify the limit and colimit respectively, which gives a further two choices. These adjectives specify the direction of the 2-morphisms involved in the (co)limit. An icon has 2-morphisms pointing towards the (co)limit, while an opicon has 2-morphisms pointing away from the (co)limit. Combining these choices, we arrive at the magic number of sixteen.
To break it down, we have:
- 3 adjectives for the cone/cocone condition: lax, pseudo, strict.
- 2 choices for the direction of 2-morphisms: icon, opicon (applies only to limits and colimits, respectively).
- 2 basic notions: limit, colimit.
This gives us 3 (laxity) * 2 (icon/opicon) * 2 (limit/colimit) = 12 variations. However, we also have the strict case, which doesn't have an icon/opicon variation, giving us an additional 2 (strict limit, strict colimit). Finally, we have the "plain" pseudo and lax versions, which also don't have an icon/opicon variation, adding another 2 (pseudo limit, pseudo colimit and lax limit, lax colimit) leading us to a grand total of 16.
Navigating the 2-Dimensional Labyrinth: Examples and Intuitions
Okay, so we've got this dizzying array of (co)limit flavors. But what do they actually mean? And when would we use one over another? Let's explore some examples and try to build some intuition.
For instance, lax limits are useful when we want to capture situations where the universal property holds only up to a non-invertible 2-morphism. This is common in situations where we're dealing with approximations or weak equivalences. Imagine trying to approximate a complex shape with simpler ones; the approximation might not be perfect, but it might be “good enough” for our purposes. Lax limits provide a way to formalize this notion of “good enough”.
Pseudo limits, on the other hand, are stricter. They require the universal property to hold up to an isomorphism. This is a stronger condition than laxity, but it still allows for some flexibility. Pseudo limits often arise in situations where we're dealing with equivalences of categories or bicategories. Think of functors that are adjoint equivalences; they're not strictly inverses of each other, but they're “close enough” for most purposes.
Strict limits are the most rigid of all. They require the universal property to hold on the nose, with equality. This is the most familiar notion of limit from 1-category theory, and it's often the easiest to work with. However, strict limits can be too restrictive in some situations, especially when dealing with higher-dimensional structures. The strictness of equality demanded by strict limits can sometimes obscure the underlying structure and relationships.
Icons and opicons further refine these notions by specifying the direction of the 2-morphisms involved. Icons have 2-morphisms pointing towards the (co)limit, while opicons have 2-morphisms pointing away. This distinction can be crucial in certain applications, especially when dealing with diagrams that have a natural directionality.
Conclusion: Embracing the Complexity of 2-Dimensional (Co)limits
So, guys, we've journeyed through the fascinating landscape of 2-dimensional (co)limits, uncovering the sixteen different flavors that arise in bicategories. It might seem overwhelming at first, but each of these notions captures a slightly different aspect of the (co)limit concept, providing us with a powerful toolkit for working with higher-dimensional structures. The key takeaway here is that bicategories provide a more nuanced and flexible framework for dealing with limits and colimits than 1-categories, allowing us to capture situations where strict equality just doesn't cut it.
By understanding the different types of (co)limits and their properties, we can gain deeper insights into the structure of bicategories and their applications in various areas of mathematics, computer science, and physics. So, next time you're faced with a complex categorical problem, remember the sixteen flavors of (co)limits, and see if one of them might be the perfect ingredient for your solution!
This exploration into the sixteen flavors of (co)limits in bicategories opens up a rich field of study. As we delve deeper into higher category theory, understanding these nuances becomes increasingly crucial. The ability to choose the right type of (co)limit for a given situation can be the key to unlocking complex structures and relationships. So, embrace the complexity, guys, and keep exploring the fascinating world of higher-dimensional category theory!