Let $\mathrm{Cell}$ be the homotopy category of finite cell complexes. The main motive of my question

Is it true that for any algebraic category $A$ there is no fully faithful functor $F: \mathrm{Cell} \to A$?

The answer may depend on what exactly is meant by algebraic categories. The following version seems to me the most natural and interesting, but comments on any other versions are welcome.

By algebraic category here I mean the category of all models of an algebraic theory, where an algebraic theory is given by the following data:

- a finite set of carriers (i.e. sorts of elements)
- a finite set of operations returning a tuple over a tuple, the scope of which is determined by a set of equalities between operations (including the empty number of equalities, of course)
- a finite set of identities imposed on operations

Example: category of small categories

- Carriers: $\mathrm{Ob}$, $\mathrm{Mor}$
- Operations: $\mathrm{dom}, \mathrm{cod}, \mathrm{id}, \circ$ (the latter is defined on those pairs of morphisms for which dom = cod)
- Identities: $$(f \circ g) \circ h = f \circ (g \circ h)$$ $$\mathrm{dom}(f \circ g) = \mathrm{dom}~f,\;\; \mathrm{cod}(f \circ g) = \mathrm{cod}~g$$ $$\mathrm{dom} (\mathrm{id}_A) = A, \;\; \mathrm{cod} (\mathrm{id}_A) = A$$ $$\mathrm{id}_{\mathrm{dom} f} \circ f = f,\;\; f \circ \mathrm{id}_{\mathrm{cod} f} = f$$

In this example, all operations returned one element, the ability to return tuples was not used (also, of course, natural operations can occur in identities: projections and direct products of morphisms). This notion differs from the finitary algebraic theory and Lover's theories. It doesn't even seem to fall into what is called generalized algebraic theory because of the possibility of operations not being everywhere defined. However, I think I've seen something like this somewhere on nlab, but I can't find it again.

P.S. Of course, it is better not to use the composition symbol (in order to use it to write identities) and write all operations in a single syntax, but in this example I could not resist and used the traditional notation.

essentially algebraic theories. For set-based models, they are also equivalent togeneralised algebraic theories. E.g., to view categories as a GAT, change “a set of morphisms” to “for any objects $x,y \in C_0$, a set $C_1(x,y)$”, and then take composition to be defined on 5-tuples $(x,y,z,f,g)$ where $x,y,z \in C_0$, $f \in C_1(x,y)$, $g \in C_1(y,z)$ (no equalities needed in this specification!) $\endgroup$homotopycategory of spaces to the category of Kan complexes, algebraic or not. $\endgroup$1more comment