nLab lax morphism

Lax morphisms

Lax morphisms


In general, if AA and BB are categories (or, more generally, any category-like things, such as objects of some 2-category) equipped with algebraic structure, a lax morphism f:ABf\colon A\to B is one which “preserves” the algebraic structure only up to a not-necessarily invertible transformation.

Of course, this transformation goes in one particular direction; a colax morphism is one where the transformation goes in the other direction. The case of 2-monads, below, provides an almost universally applicable way to decide which direction is “lax” and which is “colax”.


For algebras of 2-monads

Let TT be a 2-monad on a 2-category KK, and let AA and BB be (strict, pseudo, or even lax or colax) TT-algebras. A lax TT-morphism f:ABf\colon A\to B is a morphism in KK together with a 2-cell

TA Tf TB a b A f B\array{ T A & \overset{T f}{\to} & T B\\ ^{a} \downarrow & \swArrow & \downarrow^{b}\\ A & \underset{f}{\to} & B}

satisfying some axioms.

If the 2-cell goes in the other direction, then we say ff is a colax TT-morphism (or oplax TT-morphism). Equivalently, a colax TT-morphism is a lax T coT^{co}-morphism, where T coT^{co} is the induced 2-monad on the 2-cell dual K coK^{co} (see opposite 2-category).

If the 2-cell is invertible, we call ff a pseudo or strong TT-morphism.

For coalgebras of 2-comonads

Let WW be a 2-comonad on KK, i.e. a 2-monad on the 1-cell dual K opK^{op}, and let CC and DD be WW-coalgebras. A lax WW-morphism f:CDf\colon C\to D is a morphism in KK together with a 2-cell

C f D c d WC Wf WD\array{ C & \overset{f}{\to} & D\\ ^{c} \downarrow & \swArrow & \downarrow^{d}\\ W C & \underset{W f}{\to} & W D}

satisfying some axioms.

Note that a lax morphism of algebras for the 2-comonad WW is a colax morphism of algebras for the 2-monad W opW^{op}. The reason we choose to call this direction for coalgebras “lax” is that if TT is a 2-monad with a right adjoint T *T^*, then T *T^* automatically becomes a 2-comonad such that T *T^*-coalgebras are the same as TT-algebras, and with the above definition, lax TT-morphisms coincide with lax T *T^*-morphisms.


  • A lax monoidal functor is a lax morphism for the 2-monad on Cat whose algebras are monoidal categories. Similarly, an oplax monoidal functor is a colax morphism for this 2-monad.

  • A lax natural transformation between 2-functors CDC\to D is a lax morphism for the 2-monad on [ob(C),D][ob(C),D] whose algebras are 2-functors (which exists if DD is cocomplete and CC is small). Similarly, an oplax natural transformation is a colax morphism for this 2-monad. If DD is also complete, then this 2-monad has a right adjoint, which then as usual becomes a 2-comonad whose coalgebras are also 2-functors. The above conventions for lax morphisms between coalgebras mean that a lax natural transformation is unambiguously “lax” rather than “colax”, whether we regard the 2-functors as algebras for a 2-monad or coalgebras for a 2-comonad.

    Some authors have tried to change the traditional meanings of “lax” and “colax” in this case, but the general framework of 2-monads gives a good argument for keeping it this way (even if in this particular case, oplax transformations are more common or useful).

  • A lax functor between 2-categories is a lax morphism for the 2-monad on Cat-graphs whose algebras are 2-categories.

  • A lax algebra for a 2-monad TT is a lax morphism TA,AT\to \langle A,A\rangle for the 2-monad whose algebras are 2-monads, where A,A\langle A,A\rangle is the codensity monad of the object AA.

  • If TT is a lax-idempotent 2-monad, then (by definition) every morphism in the underlying 2-category KK between (the objects underlying) TT-algebras has a unique structure of lax TT-morphism. For instance, every functor between categories with (some class of) colimits is a lax morphism for the 2-monad which assigns those colimits; the unique lax structure map is the canonical comparison colim(FD)F(colimD)colim (F\circ D) \to F(colim D). Such a morphism is strong/pseudo exactly when it preserves the colimits in question.

  • For probability monads on a locally posetal 2-category, such as the ordered Radon monad, the lax morphisms of algebras corresponds to concave maps or a suitable generalization thereof.

Categories of lax morphisms

For any 2-monad TT, there are a 2-categories:

  • TAlg lT Alg_l of TT-algebras and lax morphisms
  • TAlg cT Alg_c of TT-algebras and colax morphisms
  • TAlg pT Alg_p (frequently written just TAlgT Alg) of TT-algebras and pseudo morphisms
  • (if TT is strict) TAlg sT Alg_s of TT-algebras and strict morphisms

We have obvious 2-functors

TAlg l TAlg s TAlg p TAlg c \array{ & & & & T Alg_l \\ & & & \nearrow\\ T Alg_s & \to & T Alg_p\\ & & & \searrow\\ & & & & T Alg_c }

which are bijective on objects, faithful on 1-cells, and locally fully faithful.

Therefore, we can also assemble a number of F-categories of TT-algebras and any suitable pair of types of TT-morphism: strict+pseudo, strict+lax, strict+colax, pseudo+lax, or pseudo+colax.

If we want to consider both lax and colax TT-morphisms together, the natural structure is a double category: there is a straightforward definition of the squares in a double category whose vertical arrows are colax TT-morphisms and whose horizontal arrows are lax ones. We could then, if we wish, add some “F-ness” to incorporate pseudo and/or strict morphisms as well.

The 2-category TAlg pT Alg_p is fairly well-behaved; for strict TT, it admits all strict PIE-limits (if the base 2-category does), and therefore all 2-limits (i.e. bilimits). When TT is accessible, TAlg pT Alg_p admits all 2-colimits as well (but not, in general, many strict 2-colimits).

However, the 2-categories TAlg lT Alg_l and TAlg cT Alg_c are not so well-behaved; they do not have many limits or colimits. But once we enhance them to F-categories, they admit all rigged limits. All three 2-categories also admit morphism classifiers?; that is, the inclusions TAlg sTAlg *T Alg_s \to T Alg_* have left 2-adjoints.


Last revised on February 2, 2024 at 00:53:30. See the history of this page for a list of all contributions to it.