Definition df-termc 49435

Description: Definition of the proper class (termcnex 49538) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49448). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 49477). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49482). See also dftermc3 49493 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 49493, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49453.

Followed directly from the definition, terminal categories are thin (termcthin 49439). The opposite category of a terminal category is "almost" itself (oppctermco 49467). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 49502).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49463) and terminal objects (dftermo4 49464). The universal properties provide an alternate proof of initoeu1 17949, termoeu1 17956, initoeu2 17954, and termoeu2 49200. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49478).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17625, 0thincg 49420, func0g 49051, 0funcg 49047, and initc 49053.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49434	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1539	. . . . . 6 class 𝑐
4		cbs 17155	. . . . . 6 class Base
5	3, 4	cfv 6499	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1539	. . . . . 6 class 𝑥
8	7	csn 4585	. . . . 5 class {𝑥}
9	5, 8	wceq 1540	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1779	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49379	. . 3 class ThinCat
12	10, 2, 11	crab 3402	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1540	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49436