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By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).

An object $A$ in a category is said to be finitely presentable (or compact) if the functor $\mathrm{Hom}(A, -)$ preserves filtered colimits.

In the (classical) $1$-truncated world, the following fact is well known: in the category of algebras of algebraic theories (aka: categories of algebras over monads of finite rank on $\mathrm{Set}$), finitely presentable objects are exactly objects having a finite presentation (given by finite generators and relations).

Is there a known generalization of this statement to the non-truncated case?

Here's what I know about it:

  1. Finitely presentable objects in $\infty\text{-}\mathrm{Groupoid}$ are exactly retracts of $\infty$-groupoids given by a finite generators and relations (see Wall's finiteness obstruction)

  2. I think that the forgetful functor will also preserve filtered colimits (but I haven't found a reference for this yet) and therefore the free one will preserve finitely presentable objects.

So,

Is it true that every finitely presentable object in a (finitary) algebraic category is a coequalizer of finitely generated free objects?

I see a discussion of (non-truncated) monads and algebraic categories in

but I haven’t noticed an answer to my question anywhere yet.

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  • $\begingroup$ What is your definition of "free" objects? Let's say, the $\infty$-category of anima (you wrote $\infty$-groupoids, but they are models of homotopy types, i.e. anima). What are free objects in your mind? $\endgroup$
    – Z. M
    Commented Feb 8 at 13:49
  • $\begingroup$ @Z.M I'm not talking about the concept of a free object in a category. Free objects are those that are in the image of the free functor (from the monadic adjunction). For the $\rm{id}$ monad on $\infty$-groupoid, the category of algebras will be $\infty$-groupoids themselves and in this case all objects are free (i.e., the same as for sets). $\endgroup$
    – Arshak Aivazian
    Commented Feb 8 at 13:56
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    $\begingroup$ @Z.M Why do you say that infinity-groupoids are models of anima? In my usual terminology, these two words are just synonyms. But, for example, Kan complexes are a model (of infinity-groupoids / anime). For example, this nlab page is full of evidence of the prevalence of such terminology . Every time I came across the word “infinity-groupoid” it meant the invariant concept itself, and not some kind of model (just like with other $(\infty, n)$-categories). $\endgroup$
    – Arshak Aivazian
    Commented Feb 8 at 14:03
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    $\begingroup$ Your question is answered in the negative by your first point: a coequalizer of fg free objects has a finite presentation, but not every compact object has a finite presentation. It is true in general that the compact objects will be retracts of those having a finite presentation (the latter is of course what "finitely presentable" should refer to...), because Ind(C)^ω is the idempotent completion of C. $\endgroup$
    – Marc Hoyois
    Commented Feb 8 at 21:10
  • $\begingroup$ @MarcHoyois Why? In the first example, all objects are free, co-equalizers are not even needed. Or do you mean there are other examples (with meaningful algebraic structure) where what you say happens? $\endgroup$
    – Arshak Aivazian
    Commented Feb 9 at 10:04

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No, this fails even for the infinity category of spaces, where "free" means "discrete". A coequalizer of free spaces is a wedge of circles, so for example $S^2$ is not of this form.

What is true is that every compact object is a retract of a $\Delta_{\leq n}^{op}$-indexed colimit of free objects, where $\Delta_{\leq n}$ is a finite full subcategory of the simplex category. Equivalently, it's in the closure of the free objects under pushouts and retracts. Note that idempotent splitting is a non-finite colimit infinity categorically. This is again illustrated in Spaces by the Wall finiteness obstruction.

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  • $\begingroup$ I don't yet understand in what sense your first paragraph is an example of the setting of my question. I am talking about finitary monads on the category of $\infty$-groupoids. A monad whose category of algebras are themselves $\infty$-groupoids, is the identity monad. And in this case, just all objects are free, aren’t they? $\endgroup$
    – Arshak Aivazian
    Commented Feb 8 at 14:46
  • $\begingroup$ On the other hand, a description of finite presentable objects as retracts of simplicial colimits of finitely generated discrete free objects would also be useful to me. Could you suggest a reference to this? $\endgroup$
    – Arshak Aivazian
    Commented Feb 8 at 15:11

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