The n-Category Café

Thank you this is very helpful. I did some digging and the result we want seems to be part of Gabriel-Ulmer duality which gives a biequivalence

$$\mathsf{Lex}^{op} \to \mathsf{LFP}$$

where $\mathsf{Lex}^{op}$ is the 2-category of categories with finite limits, finite limit preserving functors and natural transformations. $\mathsf{LFP}$ is the 2-category of locally finitely presentable categories, right adjoints between them, and natural transformations. This equivalence makes the assignment $$C \mapsto \mathsf{Lex} ( C , \mathsf{Set} )$$ and $$f \colon C \to D \mapsto (-) \circ f \colon \mathsf{Lex} ( C , \mathsf{Set} ) \to \mathsf{Lex} ( D , \mathsf{Set} )$$ i.e. composition with $f$.

Let $C$ be the category with two parallel arrows freely closed under finite limits and let $D$ be the finite limit theory for categories. Then there is a finite limit preserving functor $f \colon C \to D$ which includes $C$ into $D$ in the natural way.Then Gabriel-Ulmer duality being well-defined says that the forgetful functor this induces is a right adjoint. This is the result I want.

The most detailed description of this duality is probably in Adamek’s Algebraic Theories of Quasivarieties. Unfortunately, as far as I can tell, Adamek’s detailed description of the equivalence skips the proof that this is well defined. The original source is in German (and I can’t read it). It seems likely to me that I’m missing something.

Thanks for the (guarded) confirmation, Mike. Since Jade seems interested particularly in the case of algebras of a Lawvere theory, maybe it’s okay not to worry too much about the general locally presentable case, and just focus on the locally finitely presentable case, where the length-$n$ path construction probably gets the job done without too much fuss.

Are you sure? Even in a locally finitely presentable category, countable coproducts may not be stable under pullback, which I think is the “obvious” condition to require for that construction to work.

To be clear the original proposition for a concrete construction was to form the left adjoint as the following coproduct $$\sum_{n=0}^{\infty} E^n$$ where $E^n$ is the $n$-fold composition of the span $$V \leftarrow E \to V$$ with itself. The problem is that when you want to define composition $$\circ \colon \sum_{n=0}^{\infty} E^n \times_{V} \sum_{n=0}^{\infty} E^n \to \sum_{n=0}^{\infty} E^n$$ you want the pullback to commute with the coproducts so you can define composition on each pair $E^n \times_{V} E^m$.

To avoid this problem you say that we are in a locally finitely presentable category and compute the free category as the directed colimit you suggested. Then to define composition you still want to define it in pairs $\mathrm{Path}_n (G) \times_V \mathrm{Path}_m (G) \to \mathrm{Path}_{n+m} (G)$ so you want this pullback to commute with directed colimits. In locally finitely presentable categories is this always true? An object is called finitely presentable if its representable preserves directed colimits. It seems like this might imply that this pullback does as well but I’m not sure how.

This is just to record an example of a locally presentable category in which finite limits don’t commute with filtered colimits. Consider the $\aleph_1$-presentable category $\omega-SLat$ of posets admitting suprema of nonempty countable subsets and maps preserving those suprema. For finite posets, these are just order-preserving maps. The filtered diagrams $D_1=\{[0]\to [1]\to...\to [n]\to...\}$ and $D_2=\{[1]\to [2]\to...\to [n+1]\to...\}$, where the maps are inclusions of finite ordinals as initial segments, both have as colimit the ordinal $\omega+1$. There are two natural transformations $I,F: D_1\to D_2$ given by the inclusion of the initial and of the final segments, respectively. The equalizer of $I$ and $F$ in $\omega-SLat^{\mathbb{N}}$ is the empty diagram and has empty colimit. But $\mathrm{colim} I$ and $\mathrm{colim} F$ have nonempty equalizer, namely the single point $\omega\in \omega+1$.

It would be true only very rarely. You would need the category to be extensive for one thing, and already that’s pretty rare for a category of algebras. It would be an interesting side question to know just how rare; someone like Peter Johnstone might know the answer to that.

Was there something insufficient about the other responses you got?