nLab
preserved limit

Preservation of limits

Idea

If J:ICJ\colon I \to C is a diagram and xx is its limit in CC, then we may naïvely say that this limit is preserved by a functor F:CDF\colon C \to D if F(x)F(x) is the limit of the composite diagram IJCFDI \overset{J}\to C \overset{F}\to D. However, it is not enough to state this at the level of objects; we also need to impose some coherence conditions, preserving the entire universal cone. Furthermore, we can use a trick involving the Yoneda embedding to get a meaningful condition even if JJ has no limit in CC at all.

Definitions

Let J:ICJ\colon I \to C be a diagram and let F:CDF\colon C \to D be a functor.

Recall (see limit) that a cone over JJ in CC may be defined as an object xx of CC together with a natural transformation to JJ from the composite const x I:I!1{x}C\const^I_x\colon I \overset{!}\to \mathbf{1} \overset{\{x\}}\to C, where 1\mathbf{1} is the terminal category. Then a terminal object in the category of these cones (if it exists) is a limit of JJ in CC. Thus, a limit consists of an object xx and a natural transformation η:const x IJ\eta\colon \const^I_x \to J.

The functor FF preserves the limit (x,η)(x,\eta) if (F(x),Fη)(F(x),F\cdot\eta) is a limit of the functor FJF \circ J in DD. (Here, Fη:const F(x) IFJF\cdot\eta\colon \const^I_{F(x)} \to F \circ J is a whiskering.)

Dually, FF preserves a colimit of JJ if F op:C opD opF^\op\colon C^\op \to D^\op preserves it as a limit of J op:I opC opJ^\op\colon I^\op \to C^\op.

If FF preserves all limits or colimits of a given type (i.e. over a given category II), we simply say that FF preserves that sort of limit (e.g. FF preserves products, FF preserves equalizers, etc.).

Examples

Let II be the empty category, so that a limit of the unique functor J:ICJ\colon I \to C is a terminal object 11. Then FF preserves this terminal object if and only if F(1)F(1) is a terminal object of DD.

Let II be the discrete category 2\mathbf{2}, so that JJ picks out two objects aa and bb of CC and the limit of JJ is a product a×ba \times b of aa and bb. Note that this product comes equipped with product projections π:a×ba\pi\colon a \times b \to a and ρ:a×bb\rho\colon a \times b \to b. Then FF preserves this product if and only if F(a×b)F(a \times b) is a product of F(a)F(a) and F(b)F(b) and furthermore the product projections are F(π)F(\pi) and F(ρ)F(\rho).

A functor that preserves all small limits in CC that exist is called a continuous functor. Usually this term is only used when CC has all small limits, i.e. is a complete category.

Preservation of weighted limits

Analogously, an enriched functor between enriched categories may preserve weighted limits. Are there any tricky points that we should mention?

Preservation of limits that don't exist

Sometimes we want to say that a functor F:CDF\colon C \to D preserves a limit that does not actually exist in CC. For instance, a finitely continuous functor is usually defined as one that preserves all finite limits. If CC is a finitely complete category, then this is fine; such a functor is called left exact. But what if CC does not have all finite limits?

If CC and DD are locally small, then we can use the Yoneda lemma to turn the question into one involving categories that do have the required limits (and in fact have all limits), the presheaf categories [C op,Set][C^op,Set] and [D op,Set][D^op,Set]. (For colimits, use [C,Set][C,Set] and [D,Set][D,Set]; for VV-enriched categories, use [C op,V][C^op,V] and [D op,V][D^op,V], which will work if VV is complete.)

The left Kan extension of the composite CFDYon[D op,Set]C \overset{F}\to D \overset{Yon}\hookrightarrow [D^\op,Set] along the Yoneda embedding CYon[C op,Set]C \overset{Yon}\hookrightarrow [C^\op,Set] (which always exists) is a functor from [C op,Set][C^op,Set] to [D op,Set][D^op,Set], which may be written as F- \otimes F (alluding to the bimodule nature of profunctors). A diagram J:ICJ\colon I \to C becomes a diagram IJCYon[C op,Set]I \overset{J}\to C \overset{Yon}\hookrightarrow [C^op,Set] in [C op,Set][C^op,Set], where it has a limit. If F- \otimes F preserves this limit, then we say that FF preserves the hypothetical limit of JJ.

Since the Yoneda embedding preserves and reflects all limits, if JJ has a limit in CC, then this condition is equivalent to the condition that FF preserve it in the ordinary sense, but in general it is stronger than requiring that FF preserve the limit only if it exists in CC.

Finishing the motivating example, a flat functor may be defined as one that preserves all finite limits, whether or not they exist.

Revised on September 4, 2012 16:06:09 by Stephan Müller? (141.20.52.69)