Monomorphisms In ∞-categories

Introduction to Monomorphisms in ∞-Categories

In the realm of higher category theory, specifically within the framework of ∞-categories, the concept of a monomorphism extends the familiar notion from classical category theory. Understanding monomorphisms in this enriched setting is crucial for delving into more advanced topics like higher topos theory. This article aims to provide a comprehensive exploration of monomorphisms in ∞-categories, focusing on their definition, properties, and significance, particularly in the context of left exact functors. We will delve into the intricacies of how these morphisms behave in homotopy fibers and how they relate to the preservation properties of left exact functors, as highlighted in Higher Topos Theory (6.4.1.6).

The concept of monomorphisms in ∞-categories serves as a fundamental building block for more advanced concepts in higher category theory. Much like monomorphisms in classical category theory, which are morphisms that are left cancelable, monomorphisms in ∞-categories possess a similar characteristic, albeit in a more nuanced, homotopy-theoretic sense. The definition of a monomorphism in this context relies on the concept of homotopy fibers, which captures the higher categorical notion of the kernel of a morphism. A morphism f: X → Y in an ∞-category is considered a monomorphism if the homotopy fiber of f over any object in Y is contractible, which essentially means that the “kernel” of f is trivial in a homotopy sense. This definition, while seemingly abstract, has profound implications for the behavior of these morphisms and their role in the structure of ∞-categories.

The importance of monomorphisms in ∞-categories becomes particularly evident when considering their interaction with other categorical structures and constructions. One key area where this interaction is prominent is in the study of left exact functors. A left exact functor is a functor that preserves finite limits, which include terminal objects, pullbacks, and, importantly, kernels. In the context of ∞-categories, left exact functors play a crucial role in preserving the underlying structure and relationships between objects and morphisms. The statement that left exact functors preserve monomorphisms is a fundamental result in higher category theory, as it connects the preservation of limits with the preservation of monomorphisms. This connection is not only theoretically significant but also has practical applications in various areas of mathematics, including algebraic topology and algebraic geometry. To fully grasp the significance of this result, it is essential to understand the definition of monomorphisms in terms of homotopy fibers and how left exact functors interact with these fibers.

Defining Monomorphisms via Homotopy Fibers

To fully understand monomorphisms in ∞-categories, we must first grasp the concept of homotopy fibers. A homotopy fiber, in essence, generalizes the idea of a fiber in classical category theory, but it does so in a way that takes into account the higher homotopical structure present in ∞-categories. Given a morphism f: X → Y and an object *y: * → Y (where * represents the terminal object), the homotopy fiber of f over y, denoted as X ×Y y, is an object that represents the space of all “lifts” of y along f. More formally, it captures the universal way to complete the diagram:

 X ×Y y → X
 ↓ ↓ f
 y → Y

such that the resulting square commutes up to homotopy. This means that instead of requiring the composition X ×Y y → X → Y to be strictly equal to *y: * → Y, we only require them to be homotopic. This relaxation is crucial in the context of ∞-categories, where we often deal with objects and morphisms that are equivalent up to homotopy.

With the notion of homotopy fibers in hand, we can now define a monomorphism in an ∞-category. A morphism f: X → Y is a monomorphism if for every object *y: * → Y, the homotopy fiber X ×Y y is contractible. In simpler terms, a space is contractible if it can be continuously deformed to a single point. So, the condition that the homotopy fiber is contractible essentially means that there is a unique (up to homotopy) way to lift y along f. This definition aligns with the intuition that a monomorphism should behave like an injection, in the sense that it uniquely maps objects from X to Y. The contractibility of the homotopy fiber ensures that there is no “room” for multiple distinct lifts, thus capturing the injectivity property in a homotopy-theoretic manner.

The homotopy fiber definition of monomorphisms highlights the higher categorical nature of the concept. Unlike classical monomorphisms, which are characterized by a strict cancellation property, monomorphisms in ∞-categories are defined in terms of a homotopy condition. This means that the cancellation property holds up to homotopy, reflecting the inherent flexibility and richness of ∞-categorical structures. This definition is not just a technicality; it is essential for capturing the correct notion of injectivity in settings where morphisms are not just arrows between objects but can also be seen as objects in their own right, with higher morphisms between them. The use of homotopy fibers allows us to encode the idea that a monomorphism should uniquely determine its preimages, even when those preimages are themselves objects with a complex homotopy structure. This nuanced definition is what makes monomorphisms in ∞-categories a powerful tool for studying higher categorical phenomena.

The Significance of Left Exact Functors

Left exact functors play a crucial role in category theory, particularly in the context of ∞-categories. A functor F: C → D between two categories C and D is said to be left exact if it preserves finite limits. This means that F preserves terminal objects and pullbacks, which are fundamental constructions in category theory. In the realm of ∞-categories, where the notion of limits is generalized to homotopy limits, a left exact functor is one that preserves finite homotopy limits. This preservation property is incredibly important because it ensures that the functor respects the essential structure of the category, particularly the relationships between objects and morphisms encoded by limits.

The significance of left exact functors stems from their ability to preserve key categorical constructions. Limits, such as pullbacks and kernels, are used to define and study various properties of objects and morphisms. For example, the kernel of a morphism measures the extent to which the morphism fails to be injective, while pullbacks are used to construct fiber products, which are essential in many areas of mathematics. By preserving these limits, left exact functors ensure that these properties and constructions are not distorted when passing from one category to another. This is particularly crucial in higher category theory, where the preservation of homotopy limits ensures that the higher homotopical structure is maintained.

The preservation of finite limits by left exact functors has far-reaching consequences. One of the most important is that left exact functors preserve monomorphisms. This means that if f: X → Y is a monomorphism in an ∞-category C and F: C → D is a left exact functor, then F(f): F(X) → F(Y) is also a monomorphism in D. This result is fundamental in higher category theory because it connects the concept of monomorphisms with the preservation of limits. The fact that left exact functors preserve monomorphisms allows us to transfer information about monomorphisms from one category to another, which is essential for many applications. For instance, in Higher Topos Theory, this property is used to study the behavior of geometric morphisms, which are pairs of adjoint functors between toposes that satisfy certain left exactness conditions. The preservation of monomorphisms by the inverse image part of a geometric morphism is crucial for understanding the geometric properties of the morphism.

Proving Left Exact Functors Preserve Monomorphisms

The statement that left exact functors preserve monomorphisms is a cornerstone result in higher category theory. To understand why this is true, we need to delve into the proof, which relies on the definitions of monomorphisms and left exact functors in terms of homotopy fibers and limits. Let F: C → D be a left exact functor between ∞-categories, and let f: X → Y be a monomorphism in C. We want to show that F(f): F(X) → F(Y) is also a monomorphism in D.

Recall that f: X → Y is a monomorphism if for every object *y: * → Y, the homotopy fiber X ×Y y is contractible. To show that F(f) is a monomorphism, we need to demonstrate that for every object *d: * → F(Y) in D, the homotopy fiber F(X) ×F(Y) d is contractible. Since F is a left exact functor, it preserves finite homotopy limits, which means it preserves homotopy fibers. Therefore, we have a natural equivalence:

F(X ×Y y) ≃ F(X) ×F(Y) F(y)

where *y: * → Y is an object in C. Now, let *d: * → F(Y) be an object in D. We want to find an object *y: * → Y in C such that F(y) is equivalent to d. This is where the universal property of limits comes into play. Since F preserves limits, it maps limit diagrams in C to limit diagrams in D. Thus, if we can find a diagram in C whose limit maps to d under F, we can use the preservation property to our advantage.

Given *d: * → F(Y), consider the pullback diagram in D:

 * → F(X)
 ↓ ↓ F(f)
 d → F(Y)

Since F is left exact, this pullback is the image under F of the pullback in C:

 * → X
 ↓ ↓ f
 y → Y

where y is the object *: * → Y. The homotopy fiber X ×Y y is contractible because f is a monomorphism. Since F preserves homotopy fibers, it follows that F(X ×Y y) ≃ F(X) ×F(Y) F(y) is also contractible. But F(X) ×F(Y) F(y) is precisely the homotopy fiber of F(f) over d, which means that F(f) is a monomorphism. This completes the proof that left exact functors preserve monomorphisms in ∞-categories.

Application in Higher Topos Theory (6.4.1.6)

The result that left exact functors preserve monomorphisms has significant applications in various areas of higher mathematics, particularly in Higher Topos Theory. One specific instance where this property is crucial is in the proof of Higher Topos Theory (6.4.1.6). This result, within the context of Higher Topos Theory, deals with the properties of geometric morphisms between ∞-toposes. A geometric morphism is a pair of adjoint functors between ∞-toposes that satisfy certain left exactness conditions. Specifically, it consists of a direct image functor f: E → F and an inverse image functor f^: F → E, where f^ is left exact and left adjoint to f.

In Higher Topos Theory (6.4.1.6), the goal is to establish certain properties of these geometric morphisms, particularly concerning the behavior of monomorphisms under the inverse image functor f^. Since f^ is left exact, the result that left exact functors preserve monomorphisms directly applies. This means that if m: A → B is a monomorphism in the ∞-topos F, then f^(m): f^(A) → f^(B) is a monomorphism in the ∞-topos E. This preservation property is essential for understanding how geometric morphisms interact with the internal logic and structure of ∞-toposes.

The fact that f^* preserves monomorphisms has several important consequences in Higher Topos Theory. For example, it is used to show that the inverse image functor preserves subobjects. In an ∞-topos, a subobject of an object X is defined as a monomorphism A → X. Since f^* preserves monomorphisms, it follows that the image of a subobject under f^* is also a subobject. This is crucial for understanding how geometric properties are transported between ∞-toposes via geometric morphisms.

Furthermore, the preservation of monomorphisms by left exact functors is instrumental in establishing other fundamental results in Higher Topos Theory, such as the construction of classifying toposes and the study of localizations of ∞-categories. These results, in turn, have applications in various areas of mathematics, including algebraic topology, algebraic geometry, and logic. The interplay between left exactness, monomorphisms, and geometric morphisms highlights the deep connections between category theory, homotopy theory, and topos theory, showcasing the power and elegance of the ∞-categorical approach to mathematics.

Conclusion

The concept of monomorphisms in ∞-categories, defined through the lens of homotopy fibers, extends the familiar notion from classical category theory into a higher homotopical setting. The crucial result that left exact functors preserve monomorphisms underscores the importance of left exact functors in preserving fundamental categorical structures. This property, as evidenced by its application in Higher Topos Theory (6.4.1.6), is instrumental in understanding the behavior of geometric morphisms and the relationships between ∞-toposes. By delving into the intricacies of monomorphisms and left exact functors, we gain deeper insights into the rich and interconnected landscape of higher category theory and its applications across diverse mathematical domains.